SEISMIC STABILITY EVALUATION OF EARTH STRUCTURES

by

M. P. Singhand

M. Ghafory-Ashtiany

Final Report of the Research Supported by theNational Science Foundation under Grant

No. PFR-7823095

Department of Engineering Science & MechanicsVirginia Polytechnic Institute & State University

Blacksburg, Virginia 24061

November 1980

4. Title and Subtitle

SEISMIC STABILITY EVALUATION OF EARTH STRUCTURES

BIBLIOGRAPHIC DATASHEET 1

1. Report No.

VPI-E-80.30s. Repott Date

November 19806.

7. Author(s)

Mahendra P. SinlZh and M. Ghaforv-Ashiianv9. Performing Organization Name and Address

Virginia Polytechnic Institute andState University

12. Spoasoring Organization Name and Address

National Science FoundationWashington, D.C. 20550

15. Supplementary Notes

16. Abstracts

8. Performing Organization Rept.No.

10. Project/Task/Work Unit No.

11. Contract/Gtant No.

PFR-7823095

13. Type of Report & PeriodCovered

Final14.

An analytical approach is proposed for seismic design evaluation of earthstruc­tures like foundations and slopes. The approach is based on random vibration prin­ciples. Seismic Design inputs defined in terms of response spectra curves can bedirectly used. For verification of the proposed approach a comparative study ofthe results obtained by a rigorous non!i~~ar analysis approach and the proposedapproach is conducted •. It is observed that the proposed approach will provide asafe and conservative evaluation of an earth structure design for earthquake inducedground motions. •

17. Key Words and Document Analysis. 170. Descriptors

seismic designearth systemsfoundationsdynamic responseliquefactionearthquake engineering

17b. Identifiers/Open-Ended Terms

17c. COSAT! Field Group

18. Availability Statement

Release Unlimited

random vibrationseismic stability

19•. Security Class (ThisReport)

'TTNr, • ';:;<;": Tl'"T'I

20. Security Class (ThisPall;e

-FNCLASSIFIED

21. No. of Pages

12422. Price

".., ..... N"S-3~ \~EV ,0·73J ENDORSED BY ANSI AND UNESCO.· THIS FOR~I \IAY BE REPRODUCED USCOMM·OC 0255.1074

Acknowledgements

This investigation was sponsored by the National Science Foundation

under Grant No. PFR-7823095. The opinions, findings and conclusions or

recommendations expressed in this report are those of the authors and

do not necessarily reflect the views of the National Science Foundation.

iii

Table of Contents

CHAPTER

1. INTRODUCTION

1. 1 GENERAL. •• • • •1.2 OBJECTIVE AND SCOPE1.3 REPORT ORGANIZATION ••••.

1

134

2. ANALYTICAL FORMULATION OF DIRECT APPROACH

2.1 EQUIVALENT LINEAR EQUATIONS OF MOTION2.2 RESPONSE ANALYSIS ••••••2.3 EVALUATION OF SAFETY INDICES

2.3.1 CYCLIC DAMAGE2.3.2 FACTOR OF SAFETY

6

• • • • 6· 11· 15

. . . . . . 1620

3. SIMULATION STUDY • • 22

3.13.23.33.4

3.5

GENERAL • • • • • • • • • 22SEISMIC INPUT • • 23PARAMETERS OF SOIL SYSTEM • • • 25CONSTRUCTION OF DAMPING MATRIX • • 283. 4.1 METHOD 1 • . • • •• •• 283. 4•2 METHOD 2 • • • . • • 293. 4. 3 METHOD 3 • • • • • • • • • • 30CYCLIC DAMAGE AND FACTOR OF SAFETY EVALUATION FOR ASTRESS RESPONSE TIME HISTORY • • 32

4. NUMERICAL RESULTS · 34

4.14.24.34.44.5

4.64.7

GENERAL • • • • • • •MAXIMUM STRESS RESPONSE •FACTOR OF SAFETY EVALUATIONSPEAK FACTOR IN DIRECT APPROACH . • • •PEAK RESPONSE STATISTICS AND EQUIVALENTEARTHQUAKE DURATION • • • • . • . • . •A CONSTANT FACTOR DIRECT APPROACH • .VARIABILITY OF TIME HISTORY RESPONSE

· 34• 35

• • 37• • 38

STATIONARY• • • • 40

· 42· 43

5. SUMMARY AND CONCLUSIONS

5 01 S~Y. • • • • • • • • • • •5.2 CONCLUSIONS & RECOMMENDATIONS

REFERENCES •

TABLES •

FIGURES

iv

44

· 44· 45

47

· 52

• 58

Table of Contents (cont.)

CHAPTER Page

APPENDIX I - CORRELATION BETWEEN X (t) AND X(t) • . . • . • . . 91r s

APPENDIX II - RESPONSE EVALUATION BY SRSS FOR NONPROPORTIONALLYDAMPED SYSTEMS . . • • . • . • . • . • . 93

APPENDIX III - EXPECTED VALUES OF [G (E)X X J AND [G (E)X2J 123q r s q r

v

LIST OF TABLES

Table

1

2

3

4

5

6

Parameters of Spectral Density Functions

Standard Deviation of Relative DisplacementResponse of the Layered Media by StateVector and Normal Mode Approaches for Excita­tion Spectral Density Function of Eq. 43

Standard Deviation of the Spring Force ofthe Layered Media by State Vector and Nor­mal Mode Approaches for Excitation SpectralDensity Function of Eq. 43

Shear Modulus and S-N Curve Parameters for theLayered Media Soil

Modal Damping Ratios Obtained for System Damp­ing Matrix Constructed by Various Methods, Sec.3.4, for Acceleration Level of O.lG

Coefficient of Variation of Maximum Stress, Fac­tor of Safety, Number of Peaks per Second andBandwidth Parameter Obtained in the NonlinearApproach for Acceleration Level of O.lG

vi

52

53

54

55

56

57

LIST OF FIGURES

Figure

1. TEN LAYER SOIL STRATA 58

2. S-N CURVES FOR VARIOUS LAYERS OF FIG. 1 59

~. RAMBERG-OSGOOD STRESS-STRAIL MODEL 60

4. STRAIN DEPENDENT EQUIVALENT SHEAR MODULUS ANDDA}WING RATIOS 61

5. INTENSITY MODULATION FUNCTION 62

6. A TYPICAL SYNTHETIC ACCELERATION TIME HISTORY 63

7. ACCELERATION RESPONSE SPECTRA FOR TIME HISTORYIN FIG. 6 64

8. MEAN ACCELERATION SPECTRA FOR 1, 2, 3, 4, 5, 7,10, 15, 20, 30, 40 and 50 PERCENT DAMPING FOR 54EARTHQUAKES 65

90 MEAN RELo VELOC. SPECTRA FOR 1, 2, 3, 4, 5, 7,10, 15, 20, 30, 40 AND 50 PERCENT DAMPING FOR54 EARTHQUAKES 66

100 MEAN REL. ACCL. SPECTRA FOR 1, 2, 3, 4, 5, 7,10, 15, 20, 30 40 and 50 PERCENT DAMPING FOR54 EARTHQUAKES 67

11. ROOT MEAN SQUARE ACCELERATION RESPONSE FOR OS­CILLATOR PERIOD OF 0.2 SECS. AND DAMPING RATIO= .01 (AVERAGE OF 54 SIMULATIONS) 68

12. VARIATION OF PEAK FACTOR WITH OSCILLATOR DAMPINGAND PERIOD (BASED ON 54 ARTIFICIAL T~E HISTORIES) 69

13 0 MAXIMUM SHEAR STRESS OBTAINED BY DIRECT APPROACHAND NONLINEAR TIME HISTORY ANALYSES (54 SIMULATIONS)FOR MEAN ACCELERATION LEVEL OF OolG 70

14. MAXIMUM SHEAR STRESS OBTAINED BY DIRECT APPROACHAND NONLINEAR TIME HISTORY ANALYSES (54 SIMULATIONS)FOR MEAN ACCELERATION LEVEL OF 0.2G 71

15. MAXIMUM SHEAR STRESS OBTAINED BY DIRECT APPROACHAND NONLINEAR TIME HISTORY ANALYSES (54 SIMULATIONS)FOR MEAN ACCELERATION LEVER OF 0.4G 72

vii

LIST OF FIGURES (cont.)

Figure Page

16. FACTOR OF SAFETY OBTAINED BY DIRECT APPROACHAND NONLINEAR TIME HISTORY ANALYSES (54 SIM-ULATIONS) FOR MEAN ACCELERATION LEVEL OF O.lG 73

17. FACTOR OF SAFETY OBTAINED BY DIRECT APPROACHAND NONLINEAR TIME HISTORY ANALYSES (54 SIMU-LATIONS) FOR MEAN ACCELERATION LEVEL OF O.2G 74

18. FACTOR OF SAFETY OBTAINED BY DIRECT APPROACHAND TIME HISTORY ANALYSES (54 SIMULATIONS) FORMEAN ACCELERATION LEVEL OF q.4G 75

19. MAXIMUM SHEAR STRESS OBTAINED BY DIRECT AP-PROACH-3 FOR DIFFERENT VALUES OF PEAK FACTORSFOR MEAN ACCELERATION LEVEL OF O.lG 76

20. MAXIMUM SHEAR STRESS OBTAINED BY DIRECT AP-PROACH-3 FOR DIFFERENT VALUES OF PEAK FACTORSFOR MEAN ACCELERATION LEVEL OF O.2G 77

21- FACTOR OF SAFETY OBTAINED BY DIRECT APPROACH-3FOR DIFFERENT VALUES OF PEAK FACTORS FOR MEANACCELERATION LEVEL OF O.lG 78

22. FACTOR OF SAFETY OBTAINED BY DIRECT APPROACH-3FOR DIFFERENT VALUES OF PEAK FACTOR FOR MEANACCELERATION LEVEL OF O.2G 79

23. AVERAGE NUMBER OF PEAKS/SECOND OBTAINED BY DIRECTAPPROACH AND NONLINEAR TIME HISTORY ANALYSES (54SIMULATIONS) FOR MEAN ACCELERATION LEVEL OF O.lG 80

24. AVERAGE NUMBER OF PEAKS PER SECOND OBTAINED BYDIRECT APPROACH AND NONLINEAR TIME HISTORYANALYSES (54 SIMULATIONS) FOR MEAN ACCELERATIONLEVEL OF O.2G 81

25. AVERAGE NUMBER OF PEAKS PER SECOND OBTAINED BYDIRECT APPROACH AND NONLINEAR TIME HISTORYANALYSES (54 SIMULATIONS) FOR MEAN ACCELERATIONLEVEL OF 0.4G 82

26. BANDWIDTH PARAMETER OBTAINED BY DIRECT APPROACHAND BY TIME HISTORY ANALYSES (54 SIMULATIONS) FORMEAN ACCELERATION LEVEL OF O.lG 83

27. BANDWIDTH PARAMETER OBTAINED BY DIRECT APPROACHAND TIME HISTORY ANALYSES (54 SIMULATIONS) FORMEAN ACCELERATION LEVEL OF O.2G 84

viii

LIST OF FIGURES (cont.)

Figure

28. BANDWIDTH PARAMETER OBTAINED BY DIRECT APPROACHAND TIME HISTORY ANALYSES (54 SIMULATIONS) FORMEAN ACCELERATION LEVEL OF 0.4G 85

29. FACTOR OF SAFETY OBTAINED BY TIME HISTORY ANALYSESAND DIRECT APPROACH-3 FOR EQUIVALENT EARTHQUAKEDURATION OF 2.0 AND 1.25 SECS. FOR ACCELERATIONLEVEL OF O.lG 86

30. FACTOR OF SAFETY OBTAINED BY TIME HISTORY ANALYSESAND DIRECT APPROACH-3 FOR EQUIVALENT DURATIONS OF2.0 AND 1.25 SECS. FOR ACCELERATION LEVEL OF 0.2G 87

31. FACTOR OF SAFETY OBTAINED BY TIME HISTORY ANALYSESAND DIRECT APPROACH-3 FOR EQUIVALENT EARTHQUAKEDURATIONS OF 2.0 AND 1.25 SECS. FOR ACCELERATIONLEVEL OF 0.4G 88

32. MAXIMUM SHEAR STRESS OBTAINED BY DIRECT APPROACH-3WITH SHEAR MODULUS AND DAMPING OBTAINED FOR DIF­FERENT VALUES OF STRAIN RATIO (S.R.) -MEAN ACCEL-ERATION LEVEL = O.lG 89

33. FACTOR OF SAFETY OBTAINED BY DIRECT APPROACH-3WITH SHEAR MOD. AND DAMPING OBTAINED FOR DIFFERENTVALUES OF STRAIN RATIO (S.R.) -MEAN ACCELERATIONLEVEL = O.lG 90

ix

1. INTRODUCTION

1.1 General

Earth structures like foundation~ and slopes behave nonlinearly

when subjected to earthquake induced ground motions. Depending upon the

type of soil these structures may become unserviceable due to excessive

accumulation of strains or due to loss of strength as a result of ex­

cessive pore water pressure buildup in·a process usually called liqu~­

faction.

For accurate evaluation of a design of an earth structure for seis­

mic loads, dynamic analyses which consider nonlinear soil behavior

should be performed for a class of earthquake motions which could be

expected on the site. For design purposes, the expected earthquake

motions at a site ar~ usually characterized by response spectra curves,

such as those proposed by Housner (14), Newmark et al (29,30), Seed et

al (40) and the ATC report (2). These spectra are presumably equivalent

to an ensemble of earthquake time histories that can occur on a site,

and therefore represent a more generalized form of ,seismic design

input than any single earthquake accelerogram. Other generalized form

of seismic inputs which are collectively equivalent to an ensemble of

earthquake motions are usually defined stochastically, such as by a

spectral density function (43,52).

For earthquake inputs defined in the form of acceleration or velo­

city time histories, two different approaches are available for dynamic

analysis of nonlinear soil systems: (1) step-by-step nonlinear approach

and (2) iterative equivalent linear approach (28,36,37,41). For a

general soil system step-by-step nonlinear approach will give the most

accurate answer, but the cost of analysis can be prohibitive.

For simple one dimensional problems, however, an efficient approach

(48,49) based on the method of characteristics of partial differential

equations has been developed. A computer program called CHARSOIL has

been written for analysis by this method. Hysteretic stress-strain

behavior of soil is defined by Ramberg-Osgood Curves (33) in this pro­

gram. Further attempts have also been made to apply these methods to

some simple two dimensional problems (3,31). However, it seems that

these extensions are still in developmental stage.

In the equivalent linear approach, initially developed by Seed et

al (37) for nonlinear soil system and subsequently used in many inves­

tigations (15,16,17,26,28,35,36,41), hysteretic soil behavior is char­

acterized by equivalent and strain dependent shear moduli and damping

ratios (38); since the soil modulus and damping values are not constant

but now depend ?pon the level of strain, the analysis for a given earth~

quake motion is still nonlinear. To obtain the solution of equations

of motion with such nonlinearity, the equivalent linear approach adopts

an iterative scheme in which in each iteration a linear analysis is

performed. The solution of the equations of motion iu an iteration for

a given earthquake time history is either performed in time (16,17) or

frequency domain (26j.

To evaluate a design of a soil system by one of the above mentioned

approaches for a generalized seismic design input like response spectra,

an ensemble of earthquake time histories which are equivalent to the

generalized seismic design input must be obtained as these approaches

accept seismic input only in this form. The analyses must then be

performed for individual time histories. Also~ the analyses

2

results should be statistically processed to make a meaningfuloevalua­

tion of the soil system design. As an analysis for a single time his­

tory is computationally expensive, the design evaluation for an ensemble

of earthquake time histories becomes rather unattractive. Therefore,

the methods are being sought and developed (35,44,45) in which general­

ized seismic inputs like response spectra and spectral density function

can be directly used and the time history analyses avoided. The ap-

.proaches which can use ground response spectra directly in a design

evaluation are often called direct approaches.

1.2 Objective & Scope

The objective of this investigation is to develop and verify

such a direct approach. We have been involved in the development of

a direct approach earlier (44,45). This approach is essentially an

equivalent linear approach, but it can use a design response spectra

directly. In this investigation, this approach has been re-examined

and its analytical formulation has been refined further. Herein, it

is being proposed to use this approach for design evaluation of earth

structures.

For verification of this approach a comprehensive analytical sim­

ulation study has been conducted. In this study the dynamic response

results of a typical horizontal layered strata obtained by the direct

approach are compared with the results obtained by a step-by-step non­

linear approach. The program CHARSOIL (48) has been used in the non­

linear approach. The seismic input in the nonlinear approach consists

of an ensemble of earthquake time history, whereas that in the direct

approach it consists of a set of averaged response spectra curves

3

generated for the ensemble of time histories.

CHARSOIL program, which has been used in the simulation study,

provides a most accurate solution of a one dimensional problem. Al­

gorithms of comparable accuracy are, however, not yet available to

solve a two dimensional nonlinear problem. Therefore, in this investi­

gation, the verification of the proposed approach has been limited to

only one-dimensional problems. The formulation of the proposed ap­

proach is, however, sufficiently general for its use with two and three

dimensional problems as well.

1.3 Report Organization

Analytical development of the direct approach examined in this

study is provided in Chapter 2. Some parts of this formulation are also

available elsewhere (44) in a rather condensed form, but they have been

repeated here for the sake of continuity and completeness of the rest of

the presentation in the report.

The elements of the simulation study planned for the corroboration

of the direct approach are described in Chapter 3. This chapter in­

cludes the development of consistent seismic inputs used in the two

analysis procedures, a discussion of selected soil system parameters and

their consistency, a description of the methods used for the construc­

tion of system damping matrix used in the direct approach and the pro­

cedure for an evaluation of safety indices for a stress response time

history.

In chapter 4 the numerical results obtained by the two approaches

are presented and critically reviewed. The summary and the conclusions

are provided in chapter 5.

4

Some related analytical developments are given in the three appen­

dices. More specifically, Appendix-I provides an analytical procedure

used to estimate the correlation between nodal velocities and dis­

placements; Appendix~II, a newly developed procedure to calculate de­

sign response by SRSS for nonproportionally damped systems; and Appen­

dix-III, an analytical procedure to obtain certain expected values re­

quired in the implementation of the proposed approach.

5

2. ANALYTICAL FORMULATION OF DIRECT APPROACH

2.1 Equivalent Linear Equations of Motion

For a finite element descretization of an earth structure with

strain dependent soil modulus and damping properties, the equation of

motion at any time instant t can be written in the following form

[M]{i} + [D(S)]{~} + [S(s)]{x} = - [M]{r}X (t)g

(1)

in which [M], [D] and [S] are the mass, damping and stiffness matrices

of the structure; {x} = relative displacement vector; X (t) = the baseg

acceleration at time t; {r} = displacement influence coefficient vector

(8). The dot over a vector represents its time derivative. Thus

{x(t)} represents the relative acceleration vector. The matrices [D]

and [S] depend upon the levels of strain in various finite elements.

In the proposed direct approach it is desired to obtain the solu-

tion of Eq. 1, which has strain dependent parameters, for earthquake

excitations characterized by a spectral density function or a set of

response spectra curves. For this a quasi-linear and iterative approach

is proposed, wherein Eq. 1 is replaced by another equation in which

structural matrices do not depend upon strain, i.e.,

[M]{i}+ [C]{~} + [K]{x} = -[M]{r}X (t) (2)g

in which [C] = damping matrix and [K] = stiffness matrix, which are

assumed to be independent of the level of strain. This obviously intro-

duced an error in the equation, and also in the solution as follows:

{e} = [D(s) - C]{~} + [s(s) - K]{x} (3)

The minimization of the mean square value of the error, i.e. E[{e}Te ].

provides a mathematically convenient condition for the selection of

elemental damping ratios and shear moduli required for construction

6

of [C] and [K] matrices. In the equivalent linear time history approach

(17,37), these elemental parameters are selected corresponding to a

level of strain in the elements which usually is equal to 0.65 times the

maximum strain in the element. The factor of 0.65 represents a kind of

averaging effect which should be included in the selection of these

elemental parameters.

The minimization of mean square error as suggested here for linear-

izing the equations has been used by many researchers earlier (6,18) in

one form or another. This process is often called stochastic linear-

ization technique. For minimization of the mean square error

a~* E[{e}'{e}] = 0q

a~* E[{e}'{e}] = 0q

(4a)

(4b)

in which A* and G* are the damping ratio and shear modulus, respec-q q

tively, for the qth element. Eqs. 4 are obtained for all finite e1e-

ments. It has been shown by Iwan (18) that the solution of Eq. 4 will

minimize and not maximize the error.

Substituting for {e} from Eq. 3 and noting that [K] is independent

of A*'S and [C] is independent of G*'s the following equations are ob-q q ,

tained for the minimization of the mean square error:

E[({~}[D(S) - e]' + {x}'[S(s) - K]')a [e]{~}] 0dA* =

q

E[({~}'[D(S) - CJ' + {x}'[S(E:) - K]')d [K] {x}] 0dG* =

q

(Sa)

(5b)

or

7

E[{~}'[D(S) - C]'d

[C]{~}] + E[{x}'[S(s) - K]' d[C]{~}] o (6a)

dA* dA*=

q q

E[{x}'[S(s) - K]' a [K]{x}] + E[{~}'[D(S) - C]'a [K] {x}] o (6b)

aG* dG*=

q q

The second expected value terms in Eqs. 6a and b contain terms

which represent the cross correlation between the displacement and

velocity at the nodal points of a finite element. The correlation bet-

ween velocity and displacement at the same nodal point is of course zero

if the stochastically stationary excitation and responses are being

considered. It has also been found that the cross correlation) that is)

the expected values of the velocity at one node and displacement at the

other nodes of an element are nearly zero. (The expressions to evaluate

these correlations are given in Appendix I.) As a result) the second

terms in Eqs. 6a and b can be dropped to simplify these equations as

follows:

E[{~}' [C]' a~* [C]{~}] E[{~}' [D(s)]' d [C]{~}]= dA*q q

E[h}'[K]' d [K] {x}] E[{x}l [S(s)]' d [K] {x}]dG* = dG*q q

(7a)

(7b)

The matrices [C], [D], [K] and [S] are made up of elemental damping

and stiffness matrices. For an element these matrices can be written as

[C ] = X (c ]q q q

[D (s)] = X (s)[C ]q q q (8)

[K ] = G [K ]q q q

[5 (s)] = G (s)[K ]q q q

in which X = A*/A and G = G*/G are normalized damping and shearq q qm q q qm

modulus factors for element q; [CqJ = element damping matrix obtained

with A as damping ratio; [K ] = the stiffness matrix obtained with aqm q

8

shear modulus value of G (A and G ,respectively, may be chosen asqm qm qm

the values corresponding to high and low shear strains); A (s) = strainq

and G (s) = strain de­q

dependent damping ratio curve normalized by Aqm ;

pendent shear modulus curve normalized by G .qm

For a particular value of q, Eqs. 7a or b will have ~q

all the elements connected with the element q. Thus ~ (orq

(or G') ofq

G) valuesq

are coupled. To obtain ~ and G , Eqs. 7a and b will have to be set upq q

(9)

for all the finite elements of the system and solved simultaneously.

However, if the connectivity of the elements is ignored, Eq. 7a or b

become uncoupled which finally lead to a set of simplified expressions

to obtain ~ and G. For example, Eqs. 7a gives:q q

~ E({~ }I[C ]'[C ]{~ }) = E(A (s){~ }I[C ][C ]{~ })q q q q q q q q q q

in which {x } the displacement vector for the qth finite element. Not­q

ing that the strain s in an element depends only on the nodal displace-

ments and that the nodal velocities and displacements in an element are

only mildly correlated, the expected value on the right hand side of Eq.

9 can be written as

E[A (S){~}I[C ]'[C J{~}] = E[A (s)]E[{~}I[C ]'[C ]{~}] (10)q q q q q q

Using Eq. 10 in Eq. 9 we obtain

~ = E[A (s)]q q

(11)

That is, the most appropriate value of the damping ratio in an element

which will minimize te mean square error in Eq. 3 is the expected value

of the damping ratio random variable.

Proceeding on the same lines as above, and ignoring coupling bet-

ween various G values, Eq. 7b gives the following for the shear modulusq

factor

9

E[G (E){X }'[K ]'[K ]{x }]G = _--"-q__...:qL-._-"'q__q""----...:q___

q E[{x }'[K ]'[K ]{x }]q q q q

(12)

In neglecting the coupling between ~ and G in Eqs. 7a and b, andq q

thus Eqs. 11 and 12, we are in essence neglecting the effect of the

nodal forces applied by the adjacent element on the error minimizing

process. This, however, does not mean that the forces applied by the

adjacent elements are altogether neglected in the solution of the pro-

blem. As the complete global matrices [C] and [K] are used in the

analysis, the effect of these forces is included in the final solution.

The effect of including and disregarding these forces on ~q

and Gq

values and the final results has been investigated herein. For the

sample problem considered in this investigation, the response and fac-

tor of safety results obtained with the use of Eq. 11 and 12 were only

imperceptibly different from the results .obtined with the use of com-

plete Eqs. 7a and b. (This, however, may not always be the case for

other, more complicated soil systems (44).) Thus, here only Eqs. 11

and 12 are used.

It is noted that the solution of Eq. 2 is required to be known

to obtain X and G from Eqs. 7a and b or from Eqs. 11 and 12.q q

Specifically, the second order statistics of the response of Eq. 2 is

required before these equations can be used, For this an iterative

approach is used, in which one starts with some predecided values of Xq

and G and solves Eq. 2 to further refine these values in the nextq

iteration. The iterations are continued till a conversion in the final

values of Xq

and G is obtained.q

In this respect, therefore, this meth-

odology is similar to that being used in the equivalent linear methods

(17,36,37) .

10

In the following section, the method to obtain the solution of Eq.

2 to obtain the response statistics required in Eqs. 7a, b, 11 and 12,

and that required in the stability evaluation, for earthquake inputs

defined by spectral density function or response spectra is described.

2.2 Response Analysis

In any iteration, the syste~ is assumed to behave linearly and the

solution of equivalent linear Eq. 2 can be obtained by any suitable

technique. For an input earthquake motion defined by an acceleration

time history, step-by-step numerical integration scheme can be used, as

is done in the finite element equivalent linear techniques (16,17). For

the motion characterized by a random process, the statistics of the

response quantity can be obtained by standard random vibration proced-

ures (25,43).. For earthquake input motions defined by response spectra

curves, one has' to use the normal modes approach to solve Eq. 2 as the

spectra provide the maximum response in each mode directly.

If the ground motion can be assumed to be characterized by a spec-

tral density function, the stationary mean square value of a response

quantity can be obtained by using the normal mode procedure as follows

(43).

N N 00

E[R2] = L I YjYk~j(q)~k(q) f ~ (W)H.(w)~(w)dW (13)q j=l ~l ~g J

in which R = response quantity of interest, N = number of significantq

= relative displacement mode shape of the system; ~.(q)J

~ (w) = spectralg= jth mode shape of the response quantity of interest,

Tmodes; y. = jth mode participation factor defined as equal to {¢.} [M]{r}/J J

T{¢j} [M]{¢j}; ¢j

density function of the acceleration at the base of the system; W =

11

frequency variable; and H.(w) = relative displacement frequency transferJ

function for mode j, defined as

22·H.(w) = l/(w. - w + 2S.w.w)

J J J J(14)

in which w. = jth undamped frequency of the vibration and S. = jth modeJ J

damping ratio. An asterisk (*) as superscript on the frequency response

function denotes the complex conjugate.

The solution given by Eq. 13 is of course valid only when the equi-

valent linear matrix [C] can be diagonalized by the undamped normal

modes. That is~the matrix [C] is so-called classical type (5). Since

this matrix in any iteration is assembled* from individual elemental

matrices constructed with different damping ratios in different ele-

ments , it may not be proportional, and may have off-diagonal terms when

+pre and post multiplied by the modal matrix [~] of the syst~m. To

check what error is introduced if the off-diagonal terms of the matrix

[¢]T[C][¢] are neglected, a method has been developed to obtain the

response in the case in which these terms are retained. The theoretical

basis for this new procedure, which can also be used with the earthquake

inputs defined by response spectra curves, is given in Appendix II which

contains the copy of the paper, Ref. (46).

Using this mathematically exact procedure and the stiffness and

damping matrices obtained in the final iteration of the proposed ap-

proach for the layered soil system considered in this investigation~ Fig.

I} the displacement and elastic force response values have been obtained.

These values are shown in Table 2 and 3. Also shown are the values

obtained by the normal mode approach in which the off diagonal terms

* See Sec. 3.4+ [¢] - modal matrix

12

are neglected. The seismic input used to obtain these results is de-

fined by a spectral density function, Eq. 43, with parameters as given

in Table 1. It is seen that the difference between the two results is

rather small (in this case not more than 0.35%). Thus the off-diagonal

terms of [¢]T[C][¢] can be ignored without introducing any serious

error, and conventional modal analysis approach, Eq. 13, can be used.

All results in this report are obtained with this assumption with the

modal damping ratio S. defined as follows:J

SJ' 1 {¢.}T[C]{¢.}= 2w. J JJ

(15)

in which {¢.} are mass normalized mode shapes.J

Eq. 13 can be used to obtain the mean square response value of a

response quantity like displacement, strain, etc. in any iteration. In

evaluation of G by Eq. 12 we also need to obtain the cross covarianceq

between two nodal displacements and between displacements and strains.

These can also be similarly obtained using the following expressions

N N co

E[x x ] = I I YjYk¢j(r)¢k(s) f ~ (W)H.(W)Hi(w)dwr s g J 'j=l k=l -co

N N co

E[E X ] = 2. I YjYk¢j(r)~k(q) J ~ (w)H.(w)Hf(w)dwq r . '1 g J

J= k=l -co

(16)

(17)

in which x = relative displacement at node r, ¢.(r) = jth relativer J

modal displacement at node r, sk(q) = kth modal strain for element q.

For seismic input defined by response spectra curve, Eq. 13 can be

written in terms of given spectra values and peak factor F. (implicitly~

built in the response spectra definition) as follows:

222N 1 y.~.(q)R (w.)\'_(JJ aJ)

.L 2 4J=l F. w.J J

13

+ 2N NI I YJ'Yk~J·(q)~k(q)CTXF

j=l k=j+l(18)

in which CTXF is defined as:

2 222 2 4CTXF = {A.kR (w.) + B.kw.r R (w.)}/(F.w.)

JaJ JJ VJ JJ

(19)

and R (w.), R (w.) = pseudo-acceleration and relative velocity spectraa J v J

value at frequency w. and damping S., respectively; F. = the peak factorJ J J

of response spectra value at frequency w. and damping S.; r = w./wk ;J J J

and the factors Ajk, Bjk etc. are obtained from the solution of the

following simultaneous equations:

0 1 0Ajkl \:I

1 vk 1

2 =l:J (22)vk 1 v.r

J

1 04

0r

F., generally depend upon the frequency and damping ratio of theJ

mode, see Sec. 3.2 and Fig. 12. However, not much error is introduced

if only a single value, probably corresponding to the most dominant mode,

is used. In this investigation a single value, F, ~orresponding to the

fundamental mode of the system has been used. Also the same peak fac-

tor is used for pseudo acceleration, relative velocity and relative ac-

celeration response spectra values. With this, Eqs. 16, 17, 18 can be

defined in terms of given spectra values as follows:

E[R2] 1 (~. 2 2 ( ) 2 ( ) / 4 + 2 ~ ~ , ( )' ( ) CTX) (21)= 2" L Y.ljJ. q R w. w. f. L Y.Yk1J.!. q 1J.!k qq F j=l J J a J J j=l k=j+l J J

1 N 2 2 4E[x x ] = --2 (L y.¢.(r)¢.(s)R (w.)/w. +

r S F j=l J J J a J J

14

N NL L y·yk{¢·(r)¢k(s)

j=l k=l J Jj;l:k

(22)

1 N 2 2 4E[E x ] = --2 [L y.~.(r)~.(q)R (w.)/w.

q r F j=l J J J a J J

N N+ I L Y'Yk{~j(r)~k(q) + ~k(r)~,(q)}CTX) (23)

j=l k=l J Jk=Fj

in which CTX is defined as

22224 2CTX = {A'kR (w.) + B'kw,r R (w,)}/w, + {CJ'kRa(Wk)

JaJ JJ vJ J2 2 4

+ DjkWkRv(Wk)}/Wk

(24)

It is seen that these expressions require relative velocity re-

sponse spectra for the input motion which is, however, different from

the pseudo velocity spectra, especially in the high frequency range. If

this spectra is not available, the approximate approach suggested in

Ref. (47) can be used to obtain R (w,) from the acceleration spectrav J

values. In the simulation study, however, this spectra was generated

for its use in these expressions.

Eqs. 7a, band 12 require the evaluation of the expected values of

expressions like [G (s)x x ] and [G (s)x2]. These can be obtained asq r s q r

explained in Appendix III,

2.3 Evaluation of Stability Indices

Seismic stability of a soil system is often defined in terms of

sustained cyclic damage and factor of safety (1,24,39). The method to

be used with the proposed stochastic approach has been discussed by the

writer earlier (44,45) and is further elaborated upon in this section

for the sake of completeness of the solution of the problem.

15

2.3.1 Cyclic Damage

Soil systems lose their strength when subjected to cyclic stress

due to successive accumulation of deformation (cohesive soils) or pore

water pressure (cohesionless soils). A gradual degradation occurs in

the soil which finally renders it unserviceable for load carrying pur­

poses. The extent of degradation in soil when subjected to cyclic

stresses induced by earthquake ground motion is often measured in terms

of so-called cyclic damage (1,24,44). The concept of cyclic damage in

soils is analogous to the concept of fatigue damage in metals and often

similar procedures are used to quantify such damages for the two materials.

To assess cyclic damage (also called damage potential) the Palmgren­

Miner hypothesis is commonly used: the cyclic damage is defined as the

ratio of the applied number of cycles at a stress level to the number of

cycles the soil can take up to failure. According to this "hypothesis, the

damages due to the application of different amplitudes of stress cycles

can be linearly added to obtain the total damage. Furthermore, the

order in which different stress cycles are applied is considered immaterial.'

This latter assumption makes this hypothesis especially attractive to

use for randomly varying stresses such as those applied during an earth­

quake. A material is assumed to have failed when the total damage ap­

proaches 1.0.

It is thus seen that the calculation of damage potential for a soil

requires the stress amplitude versus the number of cycles-to-failure

plot. Such a plot is often called as S-N curve. It is obtained by

experiments on soil specimens, and depends on parameters like confining

pressure, relative density, etc. When plotted on log-log scale, an

S-N curve for a soil may be a straight line. In such a case, it may be

16

defined analytically as follows:

Nsb = C (25)

where band C are constants which depend upon the type of soil, effec-

tive confining pressure, relative density, etc. The S-N curves of var-

ious layers of the model as indicated in Table 4 are shown in Fig. 2.

The procedure to obtain cumulative damage for stochastically de-

fined stationary stress time history is given by Crandall (9) and Li~

(25) •. It is shown that the expected value of the cumulative damage is

given by

1M eoE[D] = C f

obs f (s)ds

s(26)

in which T = duration of time history, M = expected number of stress

peaks per unit time, s = stress and f (s) = the probability densitys

function of stress peaks. This probability density function is defined

as (25)

-+ as [1 + erf(sa/{o 12(1-a2)})]exp(-s2/202)

202 s ss

(27)

in which a = the band width parameter defined as = Va/M; va = zero

crossing rate of stress response; and 0 = standard deviation of stress.s

Substitution of Eq. 27 into Eq. 26 gives the following expression

for the expected value of damage

a(o 12) bTM s b+2

E[D] = C [_...::.Z-- {r(-Z-) + II} + I Z]

in which

II = feo tb/Zerf{alt/ll-aZ}exp(-t)dta

,17

(28)

(29)

.•J

and b+l---- { 2 -2} 2/1 -2 2a (I-a )

v -a s2a I2TI

s

(30)

where f(o) is the complete Gamma function.

The response due to an earthquake excitation will not be station-

ary. To obtain cyclic damage from Eq. 26, it is, however, suggested

that an equivalent stationary duration be used instead of the total ,

duration T. To obtain a conservative estimate of safety, 'the procedure

suggested by Hou (13), can be used to obtain T.

For earthquake ground excitations defined in stochastic form or by

response spectra, one can obtain the standard deviation of stress in an

element from Eqs. 13 or 21 with proper substitution for ~is. The para-

-meter q is defined in terms of the mean square response of stress and

its time derivative as follows:

2a"s

a··s

To obtain ao and a.. the following expressions can be useds s

N N co2 L L YjYk~j(q)~k(q) f w2~ (W)H.(W)~(w)dwao =s j=l k=l -co g J

N N co2 L L YjYk~j(q)~k(q) f w4~ (w)H. (w)~(w)dwa.. =:S j=l k=l -co g J

(31)

(32)

(33)

To define these quantities in terms of ground response spectra

values the following expressions can be used:

2 1 N 222 N Na.. = -2 (I y.~.(q)R (w.) + 2 L L YJ'Yk~J.(q)Sk(q)·CTWW) (35)

s F j=l J J r J j=l k=j+l

18

in which ~.(q) = jth stress mode shape, R (w.) = relative accelerationJ r J

response spectrum value at frequency w. and damping 13., and CTW andJ J

CTWW are defined as

(36)

(37)

in which A', B', C' and n' are obtained from the solution of the follow-

ing simultaneous equations:

0 1 0 1 A' (1

1 12 B' ~ -~vk v.r

J (38)2 4 =vk 1 v.r r C' [J

1 0 40 Ln0r

where vj ' vk and u are the same as defined in Eq. 20.

The relative acceLeration response spectrum value can be obtained

if such spectra curves are available; however it can also be obtained

using the relative velocity and absolute acceleration spectra values as

follows

2 = A2 + 2 2 2 2R (w.) 2(1-2B.)w.R (w.) - R (w.)r J g . J JV J a J

(39)

Evaluation of Eq. 28 requires numerical integration which can be

performed using a suitable integration technique.

Eq. 28 provides only the mean value of damage. Mark (9) has estab-

lished a method to assess the variability of damage potential. As dis-

cussed in Reference (9), the variability in damage is less if the damp-

ing involved is large. Since soil systems will tend to have a large

effective damping ratio at the stress level of any practical consequence

the magnitude of damage variability may not be significant. This is

19

substantiated by the numerical results obtained in the simulation study;

see Sec. 4.7.

2.3.2 Factor of Safety

According to Palmgren-Miner's criterion, a system becomes unser-

viceable when cumulative damage in the system reaches 1.0. However, for

any value less than 1.0, the level of cumulative damage does not provide

a direct measure of available safety margin simply because it is not a

linear measure of safety. In civil engineering practice, the term fac-

tor of safety is commonly used to measure the available safety margin.

As a result the use of this term to quantify stability in seismic load-

ing of earth structures has also been suggested (1,44).

To obtain factor of safety, some allowable stress is compared with

an "equivalent uniform stress" (1,39). Two stress time histories are

considered equivalent, if they induce the same amount of damage in a

soil element. This concept of equivalence is used to characterize a

irregular stress time history, as induced in an earthquake, by a uniform

stress time history applied for a certain number of cycles. Thus by

varying the level of stress and corresponding number of cycles, an in-

finite number of equivalent uniform stress time histories can be ob-

tained for the same amount of damage. For each such equivalent time

history with stress level S and the number of stress cycles, N , theree e

is an allowable stress S which when applied N times cyclically willa e

cause the soil to fail. The ratio

SaF.S. = s- (40)e

is called the factor of safety. For linear log-log S-N curve, this fac-

20

tor of safety is related to the damage as follows (44,45)

F.S. = n- lib (41)

It is independent of the chosen parameters, Sand N , of the equivalente e

stress time history for linear log-log S-N curve only. For nonlinear

curves, the definition of factor of safety is not as straight forward.

See Ref. (44).

21

3. SIMULATION STUDY

3.1 General

For a given set of response spectra curves, the approach described

in the previous sections can be used to assess the stability of an earth

structure. However, various assumptions have been made in formulating

the approach in a numerically tractable form. To check the validity of

the proposed approach, therefore, a numerical simulation study has been

planned. The idea is to compare the results obtained by the proposed

direct approach with the results obtained by a truly nonlinear approach

which considers the hysteretic behavior of soils under cyclic loads.

The approach developed by Streeter, Wylic and Richart (48) is a non­

linear approach which uses the method of characteristics to solve the

continuum equation of motion. The approach was originally developed for

one-dimensional problems and then extended to the two dimensional case .

by the lattice work approach'(31). A finite-element consistent two

dimensional approach has also been proposed by Ayala (3). In this inves­

tigatio~, however, only one dimensional problems have been analyzed by

the nonlinea~ approach (using the program CHARSOIL) and the proposed

direct approach.

Since the proposed approach and the nonlinear approach are quite

different from each other in as much as they use different forms of

seismic input, model the soil system differently and even use different

soil parameters, it is very essential that the problem parameters be

chosen consistently to obtain comparable results. This consistency in

various parameters chosen in the two approaches seems to have been

achieved as described in the following sections of this chapter.

22

3.2 Seismic Input

The proposed direct approach is meant to be used in design evalua-

tions and therefore has been formulated for its use with seismic input

defined by ground response spectra curves. The nonlinear approach on

the other hand uses earthquake time histories as seismic input. To make

these inputs consistent, one could choose a set of spectra curves and

obtain ground acceleration (and thus velocity) time histories by a

variety of methods that have been proposed (12,50,52). However, these

methods have some constraints such as that of enveloping a given spec-

trum by the generated time history spectrum. Some problem is also

encountered in making the generated time histories consistent with all

the spectra curves for various damping. It was, therefore, thought

better to artifica11y generate an ensemble of time histories (with

certain frequency and intensity characteristics) for their use in the

time history approach, and generate a.set of average spectra curves for

these time histories for their use in the proposed approach.

For random generation of time histories, the base accelerations are

assumed to be represented as follows:

(42)

in which X (t) a sample earthquake base acceleration; X (t) = a sto-g s

chastica1ly stationary time history motion; and e(t) = intensity modula-

tion function. X (t) is characterized by a broad-band spectral densitys

function of the following form:

3422 2

w.+4S.w.weps(w) L S. ~ ~ ~

(43)=~ 222 222i=l (wi-w ) +4l3i wi w

The parameters Si' wi and l3i of this spectral density function are given

23

in Table 1. This is a modified Kanai-Tajimi type of spectral density

function in which more terms have been added to get a broad-band effect.

The sample acceleration time history functions corresponding to this

density function are generated using the standard technique, as origin­

ally proposed by Rice (34) and subsequently used by many researchers (12,

42). Each stationary function is then further modified by an envelope

function as shown in Fig. 5. Such an envelope" function represents an~

earthquake motion somewhere between Class Band C type of motions as

classified by Jenning, Housner and Tsai (21). The choice of this func­

tion in this investigation has been made rather arbitrarily. Fig. 6

shows a sample acceleration time history record generated by this pro­

cedure and Fig. 7 shows its acceleration response spectra curves.

These artifically generated records are further modified for base line

correction to remove some inadvertent long period trends which may have

been introduced in the generation process. This correction is however

not critical as the time history response results of the soil structure

model used in this report obtained for uncorrected and corrected base

motion were only imperceptibly different from each other.

A total of 54 earthquake acceleration time histories were generated

by this procedure. (The number 54 has no special significance. It was

originally meant to be 50.) These acceleration records were integrated

to obtain the corresponding velocity time histories to be used in the

program CHARSOIL for time history generation of the results.

The acceleration time histories are used for generation of base

motion spectra curves. Absolute acceleration, relative velocity and

relative acceleration spectra curves are generated for a total of 10

damping ratios ranging from 1% to 50% of the critical. For each damp-

24

ing value, the spectra curves of the time history ensemble were statis-

tically processed to obtain the mean spectra curves. These averaged

spectra curves are used as base input in the direct approach. Figs. 8,

9 and 10 show these averaged spectra curves.

Since the proposed approach also uses a peak factor value, the time

history response of the oscillators at many spectral frequencies were

processed to obtain the mean square response time history. Fig. 11 s~ows

such a time history for an oscillator period of 0.2 sees. at a damping

of .01. Such mean square response time histories were then scanned to

obtain the maximum mean square response. The peak factor, F, is then

defined as

F = response spectrum valuemaximum root mean squared response (44)

Fig. 12 shows the variation of the peak factor with frequency and

damping. It ranges from a value of about 2.6 at some high frequency to

a value of 1.4 at very low frequencies. A value of 2. was selected for

use in the direct approach. This is the value at dominant period and

damping of the soil structu~e examined in this report. To examine the

sensitivity of the results with respect to the choice of the peak

factor, the results at other values are also reported.

3.3 Parameters of Soil System

A typical soil strata with 10 layers, 60 ft. deep and with dif-

ferent low strain shear moduli and S-N curve parameters as shown in

Table 3, has been analyzed by the two methods. For nonlinear analysis,

the stress-strain behavior of each layer is characterized by Ramberg-

Osgood curve (20,33) with following expressions:

25

Backbone Curve:

(45)

Ascending Branch

(46)

Descending Branch

(47)

in which y, T = the shear strain and stress, respectively; y ,T ,ay .y-4and R are the parameters of the model. In the analysis y = 2x10 , a

y

= 1, R = 3 have been used.

modulus GO as

T = G Yy 0 y

T is defined in terms of low strain sheary

(48)

Various branches of the curve are shown in Fig. 3.

In the proposed direct approach, however, equivalent linear pro-

perties are defined in terms of strain dependent shear modulus and damp-

ing curves (38). For a hysteretic stress strain behavior, the secant

modulus is taken as the equivalent shear modulus. Using the equation

for the backbone curve, it is seen that the secant modulus, G, at a

strain level YO' is obtained as follows:

(49)

in which TO is the shear stress corresponding to the strain level YO.

To obtain (TO/Ty) in terms of (yo/yy)' the following backbone curve

equation is solved:

26

TO R ta(-.) + (-9..)

T Ty Y

(50)

The area in a hysteresis loop represents the energy loss in a

cycle, and thus it has been related to the equivalent viscous damping

ratio as follows (28,20)

13 = --l:. !::.W4Tr W (51)

in which 13 = the damping ratio, !::.W = area of the hysteresis loop and W =

the maximum strain energy corresponding to the stress and strain levels

of TO and YO.

For Eqs. 45-47,

R-l!::.W = 4 R+l •

G -G To • (-.2.) 2 T T

G T Y YY

(52)

and if the strain energy is defined in terms of YO and TO as:

1W= 2" YOTO

1 (TO)2 GO= 2" T G

Y

(53)

the equivalent damping ratio, using Eq. 51 is obtained as follows:

13 = ~ R-l (1 _ Q....)Tr R+l GO

(54)

Equivalent shear modulus and damping ratio as defined by Eqs. 49

and 54, respectively, are used in the proposed direct approach.

Different values of these parameters are obtained if the strain

energy is defined as the area under the backbone curve. That is,

Using Eqs. 51, 52 and 55, the damping ratio S. is defined as follows:

27

Go-G

2 R-l GS = 'IT R+l --2-R--G-O---G-

1 + R+l (-G-)

(56)

Furthermore if the equivalent shear modulus G* is defined as the value

which gives the same strain energy as Eq. 55 at strain level YO' then

or

1:. G*y2 = W2 0

(57)

(58)

Eqs. 56 and 58 were also used to define the equivalent properties

in proposed direct approach. However, a better correspondence with the

nonlinear approach result was obtained when Eqs. 49 and 54 were used.

3.4 Construction of System Damping Matrix

Eq. 54 or 58 can be used to obtain the damping ratio corresponding

to a level of strain in a finite element. But, how could this damping

ratio be used to define the element damping matrix [C ] and then theq

global damping matrix [C]? Some methods that have been used or could be

used are explored in this section.

3.4.1 Method 1

In absence of any better rational procedure, the Raleigh's damping

matrix has often been used to define the damping matrix for a system.

That is,

[C] = arM] + b[k] (59)

This is a proportional damping matrix. For this matrix the damping

ratio in any mode i could be written as

2S. =~ + bw.~ w. ~

~

(60)

However, the damping ratio in a soil structure changes from element

28

to element depending upon the level of strain. To incorporate this

feature of soil systems, Idriss et al (17) proposed the use of Eq. 59

at the element level, i.e.,

[c ] = a[m ] + b[k ]q q q

To define a and b in this equation, they used Eq. 60 with

a = 6wl ' b = S/w1

(61)

(62)

in which wl = fundamental frequency of the system. The element damping

matrices, [C ], are then assembled to obtain the global damping matrixq

[C]. It is seen that if the damping ratios in various elements are the

same, system matrix [C] constructed from [C ]s will then give a classi­q

cal Raleigh damping matrix which can be diagonalized by the undamped

normal modes.

. This procedure heavily +elies on the first mode to determine a and

b, which in most cases is a dominant mode. As expected, this generally

damps out higher modes by attributing very high damping ratios to them.

(For [C] constructed by this method, the modal damping ratios obtained

using Eq. 15 were found to be unreasonably high in higher modes. See

Table 5.) Thus the response quantities which are affected by higher

modes could be in error.

3.4.2 Method 2

This is an extension of Method 1 in which constants a and bare

defined using two frequencies of the system. If the damping ratio Si

remains constant for the two frequencies, using Eq. 60 we obtain

a = 26 w.w./(w. + w.) (63)J.J J. J

b = 26/(w. + w.) (64)J. J

in which w. and w. are the two chosen frequencies. Again the choice of]. J

29

w. and w. is arbitrary. The best choice appears to be the first and the~ J

last significant frequencies. For the soil system being examined in

this investigation, the distribution of damping obtained in various

modes with w. and w. equal to the 1st and the last frequencies was more~ J

uniform. See Table 5 for modal damping values obtained in a typical

case.

An extension of this method to include more than two modal fre-

quencies to define element damping matrix was also investigated. Here,

as an extension of the method described by Wilson and Penzien (53) and

also discussed by Clough and Penzien (8) for construction of a propor~

tional damping matrix for a system, the use of the following form to

define element damping matrix was explored:

Ic ] = Im ]q q

n

l.i=O

-1 ia.(Im] Ik])~ q q

(65)

For n = 1, this equation is the same as Eq. 61. To obtain ai' a set of

simultaneous equations, as described in the text by Clough and Penzien

(8) on page 196, Eq. 13-23, were used. If these equations are used with

system IM] and Ik] matrices, a system damping matrix with desirable pro-

perties can be obtained. However, the use of this method to construct

an elemental damping matrix ICq] gave rather unrealistic modal damping

values for the system. Therefore this approach of using more than two

frequencies was not investigated any further.

3.4.3 Method 3

Another technique (4) which was investigated is based on the method

of least squares. Here we use all frequencies in Eq. 60 to estimate a

and b by minimizing the mean square error between the desired and calcu-

lated values of S.. Different weightage can also be assigned to the~

30

errors in various modes. Thus, if the weightage assigned are W., then~

the total weighted mean square error is:

e =d (66)

For minimization of ed'3ed 3ed

0aa-=ai)=

which gives rise to two simultaneous equations as:

(67)

in which

:~:J {:J (68)

N 2I w./wi ' A12 =

i=l ~

N

I Wi'i=l

N 2I W.W.,i=l ~ ~

(69)

N N\ W.S./w. and Cz = 2 \ W.S.w ..L ~ ~ ~ L ~ ~ ~

i=l i=l

In this investigation, the S. were assumed to be the same for all~

frequencies. Also, the following four different types of weighting

schemes were considered.

2 2Scheme 1: W. = (y.s.(q)R (w.)/w.)~ ~ ~ a ~ ~

Scheme 2: W. = !y.s.(q)R (w.)/w~)~ ~ ~ a ~ ~ (70)

Scheme 3: W. = Jyi si (q) I~

Scheme 4: W. = 1 (Equal weights)~

where s.(q) = strain mode shape for element q; and R (w.) = pseudo ac-~ a ~

celeration response spectra value at frequency W., Thus, the 1st~

scheme assigns weights in proportion to the contribution of a system

31

mode to the total square-root-of-the-sum-of-the-squares (SRSS) response.

Scheme 2 and 3 also assign weights on somewhat similar bases. It was,

however, observed that equal weight scheme was the one which resulted

into a more uniform distribution of modal damping ratio which were

somewhat similar to the ones obtained with the use of Method 2, dis-

cussed earlier. As discussed later, these two methods also gave the

closest response and safety factor values when compared with the time

history analysis results.

For hysteretic damping, the method of complex moduli (27,7) is

probably best suited to define the damping method for an element.

However, no formulation with such damping is available in which a given

set of base response sp~ctra can be directly used. (The frequency

domain analysis in which either a time history or a so-called spectrum

consistent spectral density function are used as in~ut have, however,

been developed (35». Thus the concept of complex moduli was not used

in this investigation.

3.5 Cyclic Damage and Factor of Safety Evaluations for aStress Response Time History

Based on the concept of equivalent uniform stress time history,

methods have developed to obtain factor of safety (1,24,39). These

methods are equivalent to the theoretical formulation developed in

Sec. 2.3 for calculation of factor of safety. However, to be completely

consistent and obtain comparable results in the direct and time history

approach, a more accurate and methodical procedure was adopted in this

investigation to obtain the factor of safety. In this procedure, the

cumulative cyclic damage is obtained by using Pa1mgren-Miner's hypo-

thesis, which is then finally converted to factor of safety using Eq.

32

41. To obtain cumulative damage, a stress time history is scanned to

locate its peaks. The damage corresponding to a peak level of S. is1.

for a S-N curve with parameters band C. Thusthen defined as = S~/c1.

total damage is equal to

1Np

(S~)D = - LC i=l 1.(71)

in which N is the number of peaks on the positive stress side. Thisp

is a discrete form of Eq. 26. Since in a stress time history, each

positive peak may not have a negative valley, the following alternate

expression may also be used

1D = 2C

NP

( Ii=l

s~ +1.

Nv

Li=l

(72)

in which N is the number of valleys (or peaks with negative stress).v

This later procedures requires scanning of a stress time history for

negative peaks also. The factor of safety is then obtained using Eq.

41. Both Eqs. 71 and 72 provided almost the same value of the factor of

safety in the numerical analysis of a typical stress time history.

33

4. NUMERICAL RESULTS

4.1 General

The numerical results have been obtained for a IO-layered strata

shown in Fig. l,with maximum shear modulus, mass density and S-N curve

parameters as defined in Table 4, by nonlinear time history analysis

approach and by the proposed direct approach. The hysteretic stress­

strain law for the soil in various layers is assumed to have been de­

fined by the Ramberg-Osgood relationships with R = 3 and a = 1, Fig. 3.

The cyclic failure rule is assumed to be described by the S-N curves

shown in Fig. 2.

The bench-mark numerical results are obtained by nonlinear time

history analyses of the strata for an ensemble of 54 artificially

generated time histories. The mean of the maximum acceleration of

these time histories was O.lG. Program CHARSOIL, with modification, was

used to obtain the nonlinear response results. The modification in

CHARSOIL were made to obtain various response quantities such as stress

time histories, zero crossing rate, peak response statistics, cyclic

damage and factor of safety at desired locations of interest in the

strata. More specifically, the maximum stress, peak response statis­

tics, cyclic damage and factor of safety values were obtained at the

center of each layer for each input time history of the earthquake

motion. Thus in all 54 sets of values were obtained for the 54 time

histories of the ensemble. These values were then processed to obtain

their mean and coefficient of variation values. The average results are

compared with the similar results obtained by the direct approach. To

further examine the validity of the proposed direct approach for higher

excitation levels, similar set of results are also obtained by time

34

history analyses and direct approach for excitation levels of 0.2g and

0.4g.

For the direct approach, the seismic input consistent with the 54

time histories used in the nonlinear approach, is defined in terms of

the psuedo acceleration, relative velocity and relative acceleration

response spectra curves. These curves are shown in Figs. 8-10 for

average maximum acceleration level of O.lg. The input curves for ac­

celeration level of 0.2G and 0.4G are obtained by appropriate scaling.

The strain dependent shear modulus and damping ratio curves, used

-in the direct approach are shown in Figs. 4. These correspond to the

Ramberg-Osgood relationships with R = 3, and a = 1. Eqs. 49 and 54

which define these curves were used in the direct approach.

Not knowing which procedure one should use to obtain the correct

damping matrix for a finite element in the direct approach, the results

with three methods, as described in Sec. 3.4, are obtained and compared.

When Method 1 of Sec. 3.4 is used to construct an elemental damping

matrix, the corresponding results are designated as "direct approach-I".

Likewise the results for the other two methods are designated as "direct

approach-2" and "direct approach-3".

In the following sections, various response results obtained by the

direct approaches are evaluated vis-a-vis the bench mark results obtained

by the time history analyses.

4.2 Maximum Stress Response

Fig. 13 shows the maximum stresses obtained at the centers of the

layers by the nonlinear analysis and by the direct approach; the results

for all three methods of damping matrix construction used in the direct

35

approach are shown in the figure. The nonlinear analysis curve in the

figure represents the average of the maximum stresses obtained for the

ensemble of input earthquake motions.

Fig. 13 shows that the results obtained by the direct approach

closely follow the stress vari~tion trend obtained in the nonlinear

approach. Among the three direct approaches, the best results (when

compared with the nonlinear analysis results) are obtained when the ele­

mental damping matrices are formed by the least squares procedures, that

is, Method 3, Sec. 3.4. The average difference in the magnitudes of

the stresses obtained by direct approach-3 and by the nonlinear approach

is about 8%, with the direct approach giving higher values. The percent

difference is largest at top. However, it is well recognized that very

near the surface the stress and safety predictions by any currently

available method are always uncertain and probably are also of little

practical significance. The stress difference at other depths, though

not large and crucial, can be attributed to: (1) inadequate modeling of

energy dissipation characteristics in the analysis; (2) choice of peak

factor values used in the analysis; (3) discretization error in the

finite element formulation used in the direct approach; and (4) certain·

soil layer property averaging procedures used in the nonlinear approach

(CHARSOIL) •

. Out of these factors, the energy dissipation modeling procedure is

probably the most important. Inadequate modeling of energy dissipation

could be due to two reasons: (1) representation of hysteretic behavior

by strain dependent damping ratio, (2) the method of formation of an

element damping matrix. The effect of the second factor seems much more

dominant, as can be seen from Fig. 13; the results are seen to depend

36

upon the method of formation of damping matrix. Although the least

square procedure used in Method 3 for this purpose seems to improve the

results, the approach itself is arbitrary. Other (probably more ra­

tional) methods are available to construct damping matrix (7,27) but

they were not implemented here. because no direct formulatio~ (the pro­

cedure in which input spectra can be used directly) is available which

can be used with such damping matrices. It is, therefore, ~esirable to

develop the direct approach further so that these later developments ~

the definition of damping matrices can also be incorporated.

Figs. 14 and 15 show similar results for higher level of excita­

tions. Comparison of the results obtained by the direct approach and

nonlinear approach again seem to establish the validity of the direct

approach for higher levels of earthquake excitations. Here again direct

approach 3 provides a better estimate of maximum stresses. As the level

of excitation increases, the nonlinear soil behavior becomes more pre­

dominant. The proposed equivalent linear method, however, seem to

handle this nonlinearity rather well. The only effect of this increased

nonlinearity seems to be that the direct method-3 now predicts somewhat

smaller values of maximum stress near the base. Whether it will do so

for all depths of strata is not verified in this investigation, but this

small order unconservative prediction of maximum stress near base should

be of little practical significance in stability evaluations of such

strata.

4.3 Factor of Safety Evaluation

The factor of safety values obtained by the direct approaches and

nonlinear analysis are shown in Fig. 16 for the mean excitation level of

O.lG. The values in the direct approach are obtained for an equivalent

37

earthquake duration of 2 secs. which is half of the strong motion phase.

(This number has been arbitrarily chosen, in absence of a better rational

basis. As discussed later, a further inspection of the time history

results seem to indicate that the equivalent effective duration is about

1.25 secs. for the earthquake motion considered in this investigation.)

It is seen that the factor of safety obtained by the direct approach are

consistently lower (a conservative prediction) than those obtained by /

the time history approach. Also the direct approach-3 in which the

damping matrix is formed by the least squares procedure gives a better

estimate of factor of safety when compared with the time history results.

The difference in the safety factor values predicted by the direct

and time history approach can again be attributed to the factors dis­

cussed in Sec. 4.2 which affected the maximum stress. The inadequate

mode~ing of energy dissipation mechanism by damping matrix, besides af­

fecting the stress response, also affects the peak response character­

istics, such as band width parameter a and thus also the factor of

safety values obtained by Eq. 28. Another factor which affects the

factor of safety values is the equivalent stationary duration T, as

discussed further in Sec. 4.5.

Similar comparisons of the factor of safety values .for higher ex­

citation levels are made in Figs. 17 and 18. Again it is seen that the

direct approaches provide a conservative estimate of seismic stability.

Also, the direct approach-3 is better than the other two approaches

considered in this investigation.

4.4 Peak Factor in Direct Approach

A suitable value of peak factor, F, is required in the direct

38

approach to obtain the mean square value of a response quantity from its

maximum value which in turn is obtained by the response spectrum analysis.

See Eqs. 21-23, 34 and 35. The value of F used in this analysis was

obtained as described in Sec. 3.2. A value corresponding to the funda­

mental frequency of the layered media was used. The effect of higher

frequencies on this value was thus ignored, though it could have been

included by using several frequency and modal damping dependent peak

factors, as in Eqs. 18 and 19.

To see what variations in the calculated maximum stress and factor

of safety values will be caused if any other value of peak factor is

used, a limited parametric study has been conducted. The results are

shown in Figs. 19 and 20 for maximum stress and Figs. 21 and 22 for

factor of safety for excitation levels of O.lG and O.2G. It is seen

that the maximum stress as well as factor of safety predictions are

indeed affected by the peak factor value in the direct approach. Thus,

use of a correct value is rather important in the prediction of response

and safety by the direct appraoch. With time history analysis results

as bench-mark the chosen value of 2.0 for the peak factor in this inves­

tigation seems to be the most appropriate value.

Various analytical procedures are available to obtain peak factor

values (19,51). Whether these methods could provide a consistent value

for the analysis performed here, was not investigated further. Since

the main purpose of this investigation was to validate the proposed

direct approach by comparing its results with the results of a truly

nonlinear analysis approach, it was important to choose a value consis­

tent with the seismic input used here to obtain comparable results.

This consistency was achieved as discussed in Sec. 3.2. However, since

39

a correct definition of this factor is important in the direct approach,

some further research effort to obtain this factor accurately for non-

stationary earthquake excitations seems necessary.

4.5 Peak Response Statistics and Equivalent Stationary EarthquakeDuration

Often a simplified formula proposed by Hou (13) has been used for

calculation of the equivalent stationary duration, T (10,11). Accordi9g

to this formula, the equivalent stationary duration for a nonstationary

input motion is approximately defined as the duration of the strong

motion phase plus 25% of the remaining earthquake duration. To obtain

the equivalent stationary duration for a response time history, a factor

which depends upon the damping value has also been suggested as follows:

C = (23S - 180.7662 + 472.86

3)

r(73)

These guidelines were initially used in this investigation to ob-

tain cyclic damage and factor of safety. It was, however, observed that

this provided a very conservative estimate of factor of safety. In fact

even a value of 2 sees. for equivalent duration provided a conservative

estimate of factor of safety in the direct approach, Figs. 16-18.

Since Eq. 28 adopted in the direct approach uses the peak response

statistics such as number of peaks/sec., M, and bandwidth parameter,

a, it is of interest to compare these quantities obtained in the two

procedures - direct approach and nonlinear time history analysis ap-

proach - to resolve the descrepancy in the peak factor results.

The number of peaks, zero crossing, and bandwidth parameter a were

then obtained for stress time histories of each layer for each of the 54

input earthquake motion time histories. The calculated values were

40

further processed to obtain mean and c.o.v. of these quantities. The

mean values are compared with the corresponding values obtained in the

direct approach.

Figs. 23 to 28 show the number of peaks/sec. and bandwidth para­

meter in various layers of the strata obtained by the three direct

approaches for the acceleration levels of 0.1, 0.2 and 0.4g. Also shown

are the average values obtained in the time history analyses. Again

direct approach-3 seems to provide a better prediction, though some

differences at higher level excitation, Fig. 25, are noted.

The results obtained in the direct approach are based on stationary

formulation, but their comparison with (nonstationary) time history

results seems to indicate that the number of peak counts, zero crossings

and thus band width parameter are not affected by"nonstationarity. This

seems plausible. The nonstationarity, however, must strongly affect the

probability density function of the peaks; that is, the density function

based on stationary assumption, Eq. 27, must have higher density values

at higher peak magnitudes than what would be obtained from a realistic

nonstationary response time history. Thus the use of Eq. 27 in Eq. 28

in the direct approach will give a higher value of damage and lower

value of factor of safety.

Since the expression for the probability density function for peaks

for nonstationary earthquake response is not available, it is not clear

as to how this conservatism in the direct approach can be removed. How­

ever, for the average maximum stress, number of peaks/unit time and the

bandwidth parameter obtained for each layer in the time history analyses,

the factor of safety values were calculated using Eqs. 28 of the direct

approach for different values of T. These values were then compared

41

with the factor of safety values obtained directly in the time history

approach. It was found that if T in Eq. is taken as 1.25 sees., the two

factor of safety values compared very well. Thus it appears that effec­

tive equivalent duration of our earthquake motions for the purpose of

calculation of factor of safety by Eq. 28 should have been 1.25 sees.

Using this new value of equivalent duration, the factor of safety

values are calculated again for the three excitations levels of O.lG, /

0.2G and 0.4G. These are shown in Figs. 29, 30 and 31. In the figures

are also shown the values obtained by the time history analysis and the

previous values for T = 2 sees. As direct approach-3 provides a better

prediction of overall response, the new values only for this approach

are shown. This reduction in the equivalent duration further narrows

the difference between the factor of safety values obtained by the

direct and time history analysis approaches. The remaining difference

which is, however, not very large, can be attributed to the factors

mentioned in Sec. 4.2.

4.6 A Constant Factor Direct Approach

In the equivalent linear time history approach (17,36,37), the

strain dependent shear modulus and damping ratio for a finite element or

layer are obtained for a level of strain equal to a fraction of the

maximum strain. This fraction, herein called as strain ratio, is

usually taken as 0.65. In the proposed direct approach also, this

procedure of choosing shear modulus and damping ratio in an iteration

can be used with significant computational simplifications if the right

value of this strain ratio is known. Figs. 32 and 33 show the maximum

stress and factor of safety values obtained for a few selected values of

42

the strain ratio ranging from 0.65 to 0.75. The results are seen to

depend upon the value of this ratio. In absence of a known value for

this ratio, the stochastic linearization technique described in the

previous sections should be used.

4.7 Variability of Time History Response

No calculations were made to obtain variability of the response

by the direct approach. Though it could be done for the maximum stress,

there are some analytical difficulties in its evaluation for factor of

safety, number of peaks and bandwidth parameter. However, the results

of simulation study provided this information. Table 6 shows the coef­

ficient of variation values obtained for the maximum stress, factor of

safety, number of peaks and bandwidth parameter obtained for the time

history results. Of special significance is the variability of factor

of safety which is of the order of 5 percent. This seems to verify the

claim made earlier in Sec. 2.3 that for soil system the variability of

damage and factor of safety is. expected to be small. Thus, the

expected value of damage and the factor of safety do provide a good

prediction of cyclic damage without much uncertainty.

43

5. Summary & Conclusions

5.1 Summary

For seismic stability evaluation of earth structures a direct

approach is presented in which seismic design inputs defined in the form

of response spectra can be directly used. The proposed approach is

similar to the equivalent linear approach in many respects, but it

does not require the design input in terms of earthquake motion time /

histories. As in the equivalent linear approach, it uses equivalent

strain dependent damping and shear modulus curves which are assumed to

be equivalent to the hysteretic stress strain behavior under cyclic

loads. Like the equivalent linear approach, the proposed approach is

also iterative in its analysis procedure; in each iteration for a given

set of stiffness and damping characteristics a linear analysis is per­

formed. The choice of shear modulus and damping values to be used in

each iteration is based on a stochastic linearization procedure, though

values corresponding to a strain equal to a constant fraction of the

maximum strain can also be used, as is done in the equ~valent linear

approach.

The analytical model of a system is supposed to be defined in terms

of mass, stiffness and damping matrices in this approach. This assumes

the decretization of a system into finite elements.

Since it is proposed to use the input response spectra directly in

an analysis, it becomes essential to use the normal mode approach. The

formulation developed herein adopts the normal mode approach and thus

all the response quantities like stress, strain, displacements, etc.,

are obtained by the (modified) square-root-of-the-sum-of-the-squares

(SRSS) procedure.

44

A SRSS procedure has also been developed in which a nonproportiona1

damping matrix can be used. However, for the example problem considered

here, it was unnecessary to use this procedure as the conventional SRSS

procedure provided very accurate results.

To find an acceptable procedure for construction of a finite ele­

ment damping matrix, three somewhat related methods have been inves­

tigated for their suitability in this work. Of these three, the one ~

based on the method of least squares is recommended.

For a typical layered soil strata, numerical results are obtained

by nonlinear time history analysis for an ensemble of 54 earthquake mo­

tions. For comparison and evaluation of the proposed approach, similar

results are also obtained by this approach for a consistent set of re~

sponse spectra curves.

Since the results of a one-dimensional problem only are examined

and compared, the following observations, in a strict sense, are appli­

cable to such earth structure problems only. However, the proposed

approach is quite general and can be used for two or three dimensional

situations as well.

5.2 Conclusions & Recommendations

An overall comparison of the results in Chapter 4 indicates that

the proposed direct approach provides a rational analytical model for a

conservative seismic stability prediction of earth structures. Seismic

input in the form of response spectra can be used. This makes the

appraoch especially suitable for evaluation of a design.

The response results obtained in the direct approach depend upon

the values of peak factor and equivalent earthquake duration used. Some

45

analytical procedures are available to define the inherent values of

peak factors built in the response spectra curves. Still some further

research effort is warranted to obtain these in a simple usuable form.

A correct evaluation of equivalent earthquake duration for its use in

the proposed direct approach is also important. A procedure which has

been available for sometime now to define equivalent earthquake duration

has been found to be rather inadequate. More research effort is re­

quired to obtain an equivalent duration which would include the nonsta­

tionarity of input, response and peak response characteristics. The

numerical results obtained for a typical horizontal strata analyzed here

indicate that the equivalent earthquake duration should be much smaller

than the strong motion phase of the input motion.

Given the correct values of peak factor and equivalent duration,

the direct approach will provide a conservative estimate of response and

seismic safety. That is, the calculated stresses will be a little

higher and the factors of safety a little lower than the values obtained

by a rigourous nonlinear approach.

The definition of a proper damping matrix in the dynamic model of a

system has always been elusive. In this investigation also, the need for

a continued development to define a better energy dissipation model in

a system by damping matrix is identified.

46

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47

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40. Seed, H. B., Maraka, R., Lysmer, J., Idriss, I. M., "Relationships Between Maximum Acceleration, Maximum Velocity, Distance fromSource and Local Site Conditions for Moderately Strong Earthquakes",Bulletin of the Seismological Society of America, Vol. 66, No.4,1976.

41. Seed, H. B. and Lysmer, J., "Soil-Structure Interaction Analysisby Finite Element Methods State-of-the-Art", Trans. of the 4thSMiRT Conference, Paper K/2/l*, Vol. K(a), San Francisco, CA, Aug.1977 •

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50

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51

Table 1: Parameters of Spectral Density Function, ¢ (w), Eq. 43g

i s. w. Si~ ~

2 rad/secft -see/rad

1 .0015 13.5 0.3925

2 .000495 23.5 0.3600

3 .000375 39.0 0.3350

52

Table 2: Standard Deviation of Relative Displacement Response of the Layered Media by StateVector and Normal Mode Approach for Excitation Spectral Density Function of Eq. 43.

ktLAIlvL Ul~IJLJ;.Lrl"i(:;H R[SPUN~E

VIW

.. 1 '"

123'tSb1bI.)

10111213141:>16111819;;::0

ROOT MLAN ~W.,~YKSSTATE VEe. fT. UNll

0.145851200+00O.14S0:>12<.tD+()l'0.1391411;#6U+OOO.1:j91~19SIJ+CUO. 12Til 2 251J+ ~)(l() Cl 12: rll2 2: 30+00O.113Z99d7U+OOO.ll::'LjOtH)2U+CJO. 'FUdJ 1 ;)(} 0-(1 t0 .. 416"181 d,U-Olo. Bl-NQ~18bD-O lO. 8 1 -'l)''! 31;)0-01(. ()~25170.:H)-01O. b52S77ZbD-(ilO. 4b :>'jlb4bD-Ol0., 4B:;':l16 tdD-O 1O.311'...2L't-ClD-CHO.:::n 7,*22 HI) D-U 1O.15"to"t6670-Cl0.1540461.>1 0-01

kQOl M£ANSQ. SQRSNNLKMAl MODE fT. UNIT

O.1't-SfJS60uO+OOO.145t1h605D+OOoe 139165 060+UO().1391b~(ibD+OO

0.12 T1Z-1400+00(\ ll..12 7 '12" 35u+f)fJ0.11 3306990+(}00.11 ))C71~fHOOO.97619061D-(H0.91 b 1 '-In, 19D-O 1.Ooofll 7313~lO-OlOoo til"' in.3 3tiO-Olo. tiS L':>26t:·fO-fn0.o52S2571iJ-Olooo .t18 5 ~101 CltU-HiO.<.tc54bl19D-Olu. 317311 ~9U-Ol0.311311990-01O.1'>"t"6i?299U-OlC.154v22 -/3(J-tJl

PERCENT [RHOR

0 .. 02I). (,,20.020.02().OlG.. CH0.01lie tll0.00v.co

-0.00-0.00-e.G1-H.Ol-O.4Jl-0.01-00001-0.2' 1-0.02-n.02

Table 3:Standard Deviation of the Spring Force Response of the Layered Media by StateVector and Normal Mode Approach for Excitation Spectral Density Function of Eq. 43.

~P~lNb ~O~~E k~SPON(E

U1~

* I *12.:?l4S67(;'J

1011121314l~1011101920

ROOT MEAN SW.,~~KSSlATt Vel.. fT ••);Nl1.

o... 1119ThO"'C4O. 4 1 1191'1 ~t tJ +IF;o. 74b1l82 JU-tU4 .o. 14.ti 11l:L~':::Li"'(;4tU.ob'74?OjjUH;<tO.66'1450lblHc<t0..., 59'7572 84U"'1l't0.5995 f3560 0'10. ~4 ..o11 b~fU r.:'"to. :>44617b'iUH.i.ltU &c 4'i 365 -({.l2 LH 00;o. 4'1 305't~t7()'"Goto. 4,:; 1bb.,.~OO",l)...O. 43"tB6411 a+(j'.0& 37t,'J~512DH) ..o. :nbI54~)'TO+C·tO. :;;, 1 :.:::>..2MIU"iAo. 312:>'-121C D+Llt0 ... 2 5 8~.1.~4b90+U..O.2~bq34b.,D"'(;"t

ROOt MEANSQ. SQRSNNORMAL MOUE fT. UNIT

0 ...12(57770+04O.4120~77t.U+O"t0 .. 14930"t95D+U.....(J.7493(;.,.4.)2U+04(j. t109912 220+040.6699121·5U+040" 5991.H7 ;JrlD+04O. 599hlb 1'';''IU+0..o. ~"J't 't 'in 7 6D ·..0...O.544'i1356U D4G. 494·112 H9U· CJ.t~ .....9 't·1C'J ltlO "Oli-

o C."i3b2..:>162u+C..O.lI31:S2:n30U+04C,,3769b9CilD .. O.,.O.37686'JiJ(JU+'J4C.j12j261.70...()~u. 31 L.:n5 05U+04{)".251H)6.<t3D... O....(i. 25 -, 16-&31u ..·J4

\

Pl:RU:Nl ERROR

C.2: l0.2.10.160.10(; .en0.07C.G'tO.Ott0.010.070.090.09O.U90.090.06(:. i'ft,

-G. (/7-o.(n-0.34-·U.3tt

Table 4: Shear Modulus and S-N Curve Parameters for the LayeredMedia Model

Layer Depth Maximum S-N Curve ParametersNo. Shear

C/sbModulus b C* = Curve No.ft. ksf e

1 6 755 5.454 0.05558 1

2 6 1400 5.454 0.05558 1

3 6 1858 5.924 0.09700 2 "

4 6 2317 5.924 0.09700 2

5 6 2786 5.924 0.09700 2

6 6 3124 5.924 0.09700 2

7 6 3444 5.924 0.09700 2

8 6 3700 5.764 0.23563 3

9 6 3995 5.764 0.23563 3

10 6 4284 5.764 0.23563 3

S = effective overburden pressuree

55

Table 5: Modal Damping Ratios Obtained for System Damping MatrixConstructed by Various Methods, Sec. 3.4. for AccelerationLevel of O.lG.

Mode Damping Rc,l.tioMethod . Method MethodNo. 1 2 3

1 .0655 .0618 .0789

2 .0964 .0319 .0406

3 .1337 .0278 .0350

4 .1596 .0266 .0333

5 .1917 .0283 .0354

6 .2496 .0337 .0422

7 .1438 .0204 .0248

8 .3123 .0395 .0495

9 .2187 .0282 .0346

10 .3718 .0445 .0559

11 .3444 .0412 .0513

12 .4275 . .0492 .0618

13 .4293 .0494 .0616

14 .4765 .0536 .0672

15 .4894 .0550 .0688

16 .5312 .0588 .0736

17 .5710 .0624 .0780

18 .6170 .0666 .0833

19 .6666 .0712 .0892

20 .7082 .0753 .0941-

56

Table 6: Coefficient of Variation of Maximum Stress, Factor of Safety,No. of Peaks per Second and Bandwidth Parameter Obtained inthe Nonlinear Approach for Acceleration Level of O.lG.

Coefficient of VariationLayer Maximum Factor of No. of Peaks/ Bandwidth

No. Depth Stress Safety Sec. Parameter

1 3 .08 .04 .09 .10

2 9 .08 .04 .10 .08

3 15 .08 .05 .13 .09

4 21 .08 .05 .13 .11

5 27 .08 .05 .16 .12

6 33 .08 .05 .16 .12

7 39 .08 .05 .12 .11

8 45 .08 .05 .14 .12

9 51 .07 .05 .12 .11

10 57 .07 .05 .13 .11

57

TOP SURFACE

ROCKBASE

. . - )I . . . , . · Layer. I. 6' .. I . . - - .

· . )

(.

~ , . .2, 6' . (.. · . . .. .. . . I.

) I. . - ..6' .

,3 ). . , .. · - (· . ... . -. · ,

6~I . 4 .· , . "

.I

. . , ·. .. · ..

6", \. · , - 5 .... .

I .. . . . .... . ·I . . . .

6'" ~. 6\ .. .. · , , . . •) ~ .t ~ . . ./

. · . .61., . · 7 ,

, · . . , (. , I . . I ..{}

. . ~ · . )- .8 I 6'· . • (. . . ., . · · . · . .

j .. .

;, I . · .. · " . 6' .· 9 ·. · . , .. , . l. ·.\ . , .( " ,

10I 6' . )• . . ·.. . . .

I . . I .•////////////////////////////////////////

c ~ Xg(t)

BASE EXCITATION

FIG. I TEN LAYER SOIL STRATA

58

1.0. . I I Iii Iii II. • i i k @ n k

CURVE 3

CURVE 2CURVE

0 0 .5-!ia::CJ)CJ)wa::l­f/)

0.2

\JI\0

0.11 2 5 10 20

NUMBER OF CYCLES50 100

FIG. 2 S-N CURVES FOR VARIOUS LAYERS OF FIG. 1

3.002.00

N~RMALIZED STRAIN

oo.......

I

0Lf)

.......

en "enUJ ALFA • 1. R 30a: .l- 0en 0

c .......UJNl-O

-Je:t~

a: 00 Lf)

z

-2.00-3.00

oLf)

.......I

FIG. 3 RAMSER ~SGOOD STRESS - STRAIN M~DEL

60

I.0 i :::::::=t I.0

O.S O.S"E E(!) DAMPING...... ~(!) RATIO en...

0 0.6 0.6 ..~

0-0:: tiCJ) a:

(J\ :::> (!)~-J::> 0.4 SHEAR MODULUS z0 RATIO 0.4 a::0 :liE:E «0::

0

«w::r: 0.2 ~ / " -10.2en

OLe=:=:: I I I dO

10-3 10-2 10-1 I 10

PERCENT SHEAR STRMN

FIG. 4 STRAIN DEPENDENT EQUIVALErIT SHEAR r/l0DULUS AND DAr~PING RATIOS

e(t)

\.0

o 2 6TIME, Sec~

15

FIG. 5 INTENSITY ENVELOPE FUNCTION

62

oIn

ACCELERATION TIME HISTORY FOR EARHtQUAKE NO = 54 DOMAXIMUM ACCELERATION IN G. UNIT = 3.241

TIME AT MAXIMUM ACCELERATION IN SEC.=4.730

6.00

FIG, 6 ATYPICAL SYNTHETIC ACCELERATION TIME HISTORY

HLLt:Lt:nH J 1 UN ::lLMLC= U.".l

TIME SCALE= 0.50

-

,~I !lr A

-.A fi n" fI" ." ~

II -

Nt !~~. V1Ji"{NW >J 12.00 14.00. 1

'-\

ol[)

('J

!

ol[)

...-<

I

.-.

.('J

ol[)

zooU:1--10

r­ITa=weD~l[)

w·C-l9uIT

0\W

ERRTHQURKE N~. = 54.0ACCEL. SPECTRA FOR 1 .2 .5 .7 .10 .15 .20 .30 .40 AND 50 PERCENT DAMPING

A.A. A

lJiVh "ANI!

tv r,..,V '\

l....-r-t I ' lA'

1/;~ II .f"'!,'-~ ,., J't\ VI

~_:::J ..., r.,..~ ...... ...."~ \ I"

~, -' /: "-

~...;;: ",r-!/ -~~/ '" iv"'---

~t\.r-- '\ ~ ic-A~

~

f'... 1\ .,r-...;.... "'-

!'iii; i'\.i\ ...........1"l~ -... f'-.. "-. I

~~ Id~l"~~~\

~, I 1

I

, I I ! I

10.0

7.0

5.0

11.0

3.0

2.0

1.11

L:l1.0

z- 0.7z0...... 0.5t-o: O.qa:~ 0.3UJuu 0.20:

0.1

0.07

0.05

O.Oq

0.03

0.02

0.010.01 0.02 0.040.06 0.1 0.2 0.3 0.5 0.7 1.0

PERIOD IN SEC.

2.0 3.0 5.0 10.0

FIG. 7 ACCELERATION RESPONSE SPECTRA FOR TIME HISTORY IN FIG. 6

64

3.0

2.0

1.0

0.7

0.5O.Y:

l:)

z 0.3-z 0.2c-l-e:a::w 0.1.....Jwu 0.07u0:

0.050.011

0.03

0.02

V/ -.-.

~ f\_/ .......- A

/ ....\\//1--' ;I ,/ .......... i'V ~ ,,,

.--'/

",-~~ r;::~ :--.i' .. ..~ ,~... V\

~E;~

r-i'-:-- i'.

~tl-\~- r-

~'- r-r-. !-"h:- ~......

~ :'-- l'\.N .'\;'

""'II ,'\l"~ ~~~~ ~,.....

""llllil

~~~~

0.010.02 0.01l0.06 0.1 0.2 0.3 0.50.7 1.0

PERIOD IN SEC.2.0 3.0 5.0

FIG.8 - MEAN ACCLERATION SPECTRA FOR 1.2.3.1.5.7.1015 . 20 . 30 . 10 AND 50 PERCENT DAMPING FOR 54 EARTHQUAKES

65

IA( ,\ -'"

I" ',I 'v \. V ~ilj~ ~ ~

1/ ~ f'=: :~

VrV~ I- :-, ":---8::~.-/ V ~ --;/ V V V

10"'"""" .- .-- - v ....~~r 11'1,;' 1--- I.--- ~ ~--V/ ./ '/ "

10' V- I-" i-"'" I-.- ". .... ;"".-

il V V~ / ./ )/ , ~ ......... ...~ ....

1// V~~/ ....... /~ :;:;1' ....

I II '/ V

IV~~~V/ ~V V"V// / '/

IIfN' 1/

III r/, I /1/'1 /

/ I Jif. 'I. 'j

/ '/ 'It 'I 1//1

/1 /J. ~V~V-I 'J I 'l/ ill I / ,J 'I

h~ 'I '/,

/J~~,/,

tIIA 'fl

"II.

~liDV

I---

,

2.0

1.I!

1.0

0.7

0.5

O.I!

0.3

0.2

• 0.1uwU') O. 07........t- 0.05lJ,.

0.011z..... 0.03

>-t-..... 0.02uo-lW>

0.01

O. 01

0.010.00

0.00

0.00

0.000.02 0.01l0.06 0.1 0.2 0.3 0.5 0.7 1.0

PEAI~D IN SEC.2.0 3.0 5.0

FIG.S -MEAN REL.VELOC.SPECTRA FOR 1.2.3.1.5.7.10.15.15 .20 .30 .10 AND 50 PERCENT DAMPING FOR S1 EARTHQUAKES

66

// "-.. r--... '\----,/ / I\Y,,~

/ "..:- - I'.. I~~

II/~V.... ~ ~ f\}':

V I ... ,..... ~

/,/

V II~/ f.-'V ....

~VI-' f.-'V- r-..V III -,/ I--- --/ .... -,/ V ~ i/ l.- ..... iJo,/ l/ VIII ......

1/ ./ 1/ ;I' ~

VI[/ vII ;I'

IJ VJ / / V/t/

~~/ //~V/ !/

- '1/ / /1

IIij/

3.0

2.0

1.0

0.7

0.5O.q

t::)

z 0.3.....z 0.210......t­o:IX:IJ.J O. 1-lI.LJu 0.07u0:

0.050.01i

0.03

0.02

0.010.02 0.01l0.06 0.1 0.2 0.3 0.5 0.7 1.0

PERIOD IN SEC.2.0 3.0 5.0

FIG. 10 - MEAN REL. ACCL. SPECTRA FOR 1.2.3. "'t • 5 . 7 . 10 .15 .20 .30 . "'to AND 50 PERCENT DAMPING FOR 54 EARTHQUAKES

67

oo.......

EART HQURKE NrJ. = 54. 00CD

·0If)

f-I--l

Z:::Jo

(D

·GO

ZI--l

Wo=:J=t'

0'1.....J~<XlIT>(f)

Lo(£ru

·0

oo

0 0 . 00

DAMPING= 0.010PERIOD= 0.200RMS SCALE= 0.2000

'~

I _." -,. I --------\' I I I ---.- I

2.00 4.00 6.00 8.00 10.00 12.00 14.00 16.00TIME IN SECD

FIG. II ROOT MEAN SQUARE ACCELERATION RESPONSE TIME HISTORY FOR OSCIllATOR

PERIOD OF 0.2 SECS. AND DAMPING RATIO =' .01 (AVERAGE OF 54 SiMULATIONS)

3.0 I I0.0

0:: 0.05 0.50

g 0.19...= "'"~0.20o ~~ 010il 2.0 0.20J 0:05"", ~ 0.50 " 0.01~ <t

W0-

I1.0

0' I! 1111111 I 1"11111 ! , I

0.01 0.02 0.05 0.1 0.2 0.5 1.0 20 50 10.0

PERIODS, SEes.

FIG. 12 VARIATION OF PEAK FACTOR WITH OSCILLATOR DAMPING AND PERIOD

(BASED ON 54 ARTIFICIAL TIME HISTORIES)

TOP LAYER NO.0

Ag =OJ G,~

10 2

3

~ 20 4w NONLINEAR ANA.wLL

Z 5:J: 30~a.. 6w DIRECT APR -30

40 DIRECT APR -I7

DIRECT APP. - 28

50

1000800600400

BASE

200

60 ~,..,......_---l. --'- -'-- ..L..--__----.l._~

aMAXIMUM STRESS, PSF

FIG. 13 MAXIMUM SHEAR STRESS OBTAINED BY DIRECT APPROACH

AND NONLINEAR TIME HISTORY ANALYSES (54 SIMULATIONS) FORf'1EAN ACCELERATION LEVEL OF O.lG

70

~. Ag =0.2 G 1

~, .~

2- ~~,

""~ 3"~~ ,

~~'"' NONLINEAR

~ .4ANA. ~~DIRECT APR -I

. ~'~5

DIRECT APR -3 ~~"\.,\'\

6DIRECT ' ~APP. -2 ""\. ~

. ~~\. 7

\'\ 8

I- \\\ 9

\ \\\\ 10

BASE. . .////

LAYER NO.o

10

20t-WWI.L.

Z 30:I:t-a..w0

40

50

60o

TOP

250 500 750 1000MAXIMUM STRESS, PSF

1250 1500

FIG. 14 MAXIMUM SHEAR STRESS OBTAINED BY DIRECT APPROACHAND [~ONLINEAR TU1E HISTORY ANALYSES (54 SIf~ULATIONS) FORMEAN ACCELERATION LEVEL OF O.2G

71

TOPOr-----------------------.

10

20....IJJIJJLL

Z 30.:r:....

DIRECT "APP. -2a..IJJ0

40

50

2000800 1200 1600

MAXIMUM STRESS, PSF

BASE60 L,-,-,..,...-_--L. -J-. "'--__---'- -J-.-.J

o 400

FIG. 15 MAXIMUM SHEAR STRESS OBTAINED BY DIRECT APPROACHAND NONLINEAR TIME HISTORY ANALYSES (54 SI~ULATIONS) FOR

MEAN ACCELERATION LEVEL OF O.4G

72

TOP0

\\\TIME HISTORY AVG.,\

'\10 ---- DIRECT APR-I

~\ --- DIRECT APP. 2

\\ - - DIRECT APP. 3

20 \ \~.... \ \\

LLJ \\\w Ag = O. t GlJ-

Z30 \\

:::c~\....

a..w \\ \0

40 \\'\~'':\'\\

50 ,\ \\\\

,\ \\\

BASE60

0 \.0 2.0 3.0 4.0 5.0FACTOR OF SAFETY

FIG. 16 FACTOR OF SAFETY OBTAINED BY DIRECT APPROACH ANDNONLIHEAR TIME HISTORY ANALYSIS (54 SIMULATIONS) FOR MEANACCELERATION LEVEL OF O.lG

73

TOP0..--------------......,

10

20

I­WWI.L

Z 30J:I­a.wo

40

50

-- TIME HISTORY AVG.----- DIRECT APP. -I--- DIRECT APP. -2--- DIRECT APR -3

BASE

4.01.0 2.0 3.0FACTOR OF SAFETY

60 s.,.,...,~---'-------l.---~----J

o

FIG. 17 FACTOR OF SAFETY OBTAINED BY DIRECT APPROACHAim NONLINEAR TIME HISTORY ANALYSES (54 SIMULATIONS) FORMEAN ACCELERATION LEVEL OF 0.2G

74

TOP0

\\\

~ TIME HISTORY AVG.10----- DIRECT AP~ -I

\ --- DIRECT APR -2

~ --- DIRECT APP. -3

20 '~.... \, Ag =0.4GLIJLIJ ~~~

z 30

\:I:....a.LIJ

\~C

40 \~\

~•50~

\

\BASE

600 1.0 2.0 3.0 4.0

FACTOR OF SAFETY

FIG. 18 FACTOR OF SAFETY OBTAINED BY DIRECT APPROACH ANDTIME HISTORY ANALYSES (54 SIf1ULATIONS) FOR MEAN ACCELERATIONLEVEL OF 0.4G

75

TOP0-----------------------,

~"'--~F: = 2.2

~-4--RF:=2.0

~F: =2.5RF. =ta-~Iorto\.

RF. = 1.5 -.-.

TIME HISTORY AVG.-~"",\

10

20

40

50

....LaJLaJLa..Z 30-~LaJo

1000400 600 800MAXIMUM STRESS, PSF

BASE60 L.,..,..~.=.=..--L...-------JI....---...J-------l---~~

o 200

FIG. 19 MAXIMUM SHEAR STRESS OBTAINED BY DIRECT APPROACH-3FOR DIFFERENT VALUES OF PEAK FACTORS FOR MEAN ACCELERATIONLEVEL OF 0 I IG

76

TOPO~---------------------,

~~-RF: =2.0~-RF. =2.2

"W---P.F. =2.5

P.F.=1.5RF. =1.8~~

TIME HISTORY AVG.--4---~

BASE60 ~";::'''':'::'':::....J... ..L.....__--J. ...I--_~----Iu.....A.--a_J-I

o 250 500 750 1000 1250 1500MAXIMUM STRESS, PSF

10

40

50

20

....WlJJu..~ 30::J:

h:wo

FIG. 20 r1AXIi'1UM SHEAR STRESS OBTAINED BY DIRECT APPROACH-3

FOR DIFFERENT VALUESOF PEAK FACTORS FOR r1EArJ ACCELERATION

LEVEL OF 0.2G

77

TOPOr-----------------.Ag =0.1 G

\. TIME HISTORY AVG.

\4+-+- RF. = 2.2W-;-- R F. = 2.5

\,\\\\

\ ,

p.F. =1.5 -~P. F. = l.8 -----It-l

p.t=: =2.0--\-~

20

40

50

10

....lJJlJJLL

~ 30J:....a..lJJo

BASE

5.04.01.0 2.0 3.0FACTOR OF SAFETY

60 I..rr~---I----....I------.L---"""'-----'"

o

FIG. 21 FACTOR OF SAFETY OBTAI~ED BY DIRECT APPROACH-3

FOR DIFFERENT VALUES OF PEAK FACTORS FOR MEAN ACCELERATIONLEVEL OF 0 I 1G

78

TOP0..---------------...,

10

20.-IJJlLJu..

~ 30:z:I­a.wo

40

50

,,\,\ Ag =0.2G

\~·tE-TIME HISTORY AVG.\

w--RF. = 2.5\~- P. F. = 2.2HrH---P'F. = 2.0

PF: =1.8-~p.F. =1.5~

BASE

4.01.0 2.0 3.0FACTOR OF SAFETY

60 ~,..,,---.....I--__L-------~

o

FIG. 22 FACTOR OF SAFETY OBTAINED BY DIRECT APPROACH-3

FOR DIFFERENT VALUES OF PEAK FACTOR FOR MEAN ACCELERATIONLEVEL OF O.2G

79

TOPOr----------------------.

-- TIME HISTORY AVG.- - - _. DIRECT APP. -I--- DIRECT APR -2~ - DIRECT APP. - 3

10

20I-WWu..z- 30:t:I­a.wo

40

50

//

"//

/I

II

II

II

II

II

I,III

• IIIII\,,I\I,,\,\\

BASE

124 6 8 10NUMBER OF PEAKS/SEC.

60 ~.,..y----'----..a..----J.---..........----'2

FIG. 23 AVERAGE NUf1BER OF PEAKS/SECOND OBTAINED BY DIRECTAPPROACH Ai'm NONLINEAR TIME HISTORY ANALYSES (54 SH1ULATIONS)

FOR MEAN ACCELERATION LEVEL OF O.lG

80

TOP0

I/

//

/

to II

IAg = 0.2 GI

II

I20 I

Il- I

LU IW I TIME HISTORY AVG.LL I

I -- - - DIRECT APR -Iz 30,

--- DIRECT APR-2I

::I:,

--- DIRECT APR-3I- r0- fW ra ,,

40,\,,

III

50 II\,,\

BASE60

2 4 6 8 10 12NUMBER OF PEAKS/SEC.

FIG. 24 AVERAGE NUMBER OF PEAKS PER SECONDS OBTAINEDBY DIRECT APPROACH AND NONLINEAR TIME HISTORYANALYSES (54 SIr1ULATIONS) FOR f1EAN ACCELERA­TION LEVEL OF 0.2G

81

TOP0,----------------------,

12.0

-- TIME HISTORY AVG.--- - -- DIRECT AP~ -I--- DIRECT AP~ -2-- DIRECT AP~ -3

4.0 6.0 8.0 10.0NUMBER OF PEAKS/ SEC.

60 L.,...,..,r-r--_--L. .J..-.__--L. ....L.-__---'

2.0

50

40

20

10

f­IJJlJJlJ..

Z

:I: 30f-a..LLIo

FIG. 25 AVERAGE NUMBER OF PEAKS PER SECONDS OBTAINED

BY DIRECT APPROACH AND NONLINEAR TIf1E HISTORY

ANALYSES (54 SIf1ULATIONS) FOR r'1EAN ACCELERATION

LEVEL OF O.4G

82

TOP0....----------------,

A =0.1 G10

I

20

"....

TIME HISTORY '\lLI

\\w ----- DIRECT APR -II.L

z --- DIRECT APR -2 I \

30 - - DIRECT AP~ -3 l \:r:t-o. \ \lLI \\0

40 . II

~1\

50 1\'Ij

BASE60

0 0.5 1.0BANDWIDTH PARAMETER 1 a

FIG. 26 BANDWIDTH PARAMETER OBTAINED BY DIRECTAPPROACH AND BY TIME HISTORY ANALYSES(54 SIMULATIONS) FOR MEAN ACCELERATIONLEVEL OF O.lG

83

TOP0,.------------------,

\\~\ "\

\\\\\\\\\\\\\)II,I\

\\

\)

II

I

/IIIIIIIII

Ag = 0.2 G10

50

20

40

ffi TIME HISTORY "z ------ DIRECT APR -I j

30 --- DIRECT APR -2J: -- DIRECT APR -:3t- I

~ \I'I'~tj!

BASE60 ~r-r--------L.--------.I

o 0.5 1.0BANDWIDTH PARAMETER, a

FIG. 27 BAND WIDTH PARAMETER OBTAINED BY DIRECTAPPROACH AND TIME HISTORY ANALYSES (54

SIMULATIONS) FOR MEAN ACCELERATION LEVELOF 0.2G

84

TOP0,.----------------,

10

1.0

\

\IIII'" '" ...

'" '" '" ..... ...I

//

II\

\\

\\

//

//,,

\\\\I

II

II

BASE

0.25 0.5 0.75BANDWIDTH PARAMETER, a

--TIME HISTORY-----DIRECT APP. -I I

20 -- - DIRECT AP~ -2- -DIRECT APP. -3 1/

I,

II~~

\'

J

II/ ~

50

60 ~.,...".-._---'- -'-- .L...-.__--1

o

30

40

f­WWI.L

Z

::r::f­a.wo

FIG. 28 BAND HIDTH PARM1ETER OBTAINED BY DIRECT

APPROACH AND TIME HISTORY ANALYSES (54

SIMULATIONS) FOR MEAN ACCELERATION LEVEL

Of 0.4G

85

TOPOr-----------------....,

10

20t­w.WlL.

Z- 30:I:.....a.wo

40

50

BASE

TIME HISTORY AVG.

M-+--FOR T= 1.25 SEes.

~+---FOR T = 2.0 SECS.

60 '-77-~-~---.A------I----.A----......o 1.0 2.0 3.0 4.0 5.0

FACTOR OF SAFETY

FIG, 29 FACTOR OF SAFETY OBTAINED BY TIME HISTORYANALYSES AND DIRECT APPROACH-3 FOR EQUIVA­LENT EARTHQUAKE DURATION OF 2,0 AND 1,25SECS. FOR ACCELERATION LEVEL OF O.lG

86

TOP0..---------------------,

10

20t­IJJIJJIJ..

~ 30:I:I­0..IJJo

40

50

BASE

Ag = 0.2 G

TIME HISTORY AVG.

~-FOR T = 1.25 SECS.

kt-+-FOR T= 2.0 SECS.

1.0 2.0 3.0 4.0FACTOR OF SAFETY

5.0

FIG, 30 FACTOR OF SAFETY OBTAINED BY TIME HISTORYANALYSES AND DIRECT APPROACH-3 FOR EQUIVA­LENT DURATIONS OF 2,0 AND 1.25 SECS. FORACCELERATION LEVEL OF O,2G

87

TOP0-----------------,

10

20I­WWlL,

Z- 30::cl-e.wo

40

50

TIME HISTORY AVG.

......--- FOR T= 1.25 SECS.

,.&+-- FOR T = 2.0 SECS.

4.0

BASE60 L...,..,..~---L.---....L----&...------&

o 1.0 2.0 3.0FACTOR OF SAFETY

FIG. 31 FACTOR OF SAFETY OBTAINED BY TIME HISTORYANALYSES AND DIRECT APPROACH-3 FOR EQUIVA­LENT EARTHQUAKE DURATIONS OF 2.0 AND 1.25SECS. FOR ACCELERATION LEVEL OF 0.46

88

TOP0..---------------------,

10

20

t-WWu..

~ 30I:t-o. TIME HISTORY AVG.w0

40

50

1000

BASE

200 400 600 800MAXIMUM STRESS, PSF

60 l.,-,-,....,..-_--L- ..J.-__-L -'--__--J

o

FIG, 32 MAXIMUM SHEAR STRESS OBTAINED BY DIRECTAPPROACH-3 WITH SHEAR MODULUS AND DAMPINGOBTAINED FOR DIFFERENT VALUES OF STRAINRATIO (S.R.) - MEAN ACCLERATION LEVEL = O,IG

89

TOPO~---------------,

10 Ag=O.1 G

TIME HISTORY AVG.

20t-LLJLLJLL.

Z30

:I:t-o.LLJ0

40

50

BASE

4.01.0 2.0 3.0FACTOR OF SAFETY

60 ~~_-.a... .l...-__........J--__----'

o

FIG. 33 FACTOR OF SAFETY OBTAINED BY DIRECT APPROACH-3

WITH SHEAR MOD, AND DAMPING OBTAINED FOR DIF­FERENT VALUES OF STRAIN RATIO (S,R.) - MEANACCELERATION LEVEL = O.lG

90

APPENDIX - I

Correlation Between x (t) and ~ (t):r s

To evaluate the effect of correlation between displacement at one

node and velocity at another on Eqs. 6a,b, the following equations were

used to obtain the magnitude of correlation coefficient.

Correlation Coefficient:

E[x (t)X (t)]r sa a­x xr s

(1.1)

E[x (t)~ (t)] = -2r s

co

I_co ¢g(W) [WjWk(SkWj-SjWk)W2 - (WkSk-WjSj)w4]

2 2IHj

(w) I 11\(w) I dw (1.2)

The derivation of Eq. 1.2 assumes a stationary response. For a

given spectral density function, this equation can be integrated by a

suitable integration procedures. To express it in terms of base re-

sponse spectra values, the following equation can be used:

E[x (t)~ (t)]r s

(1.3)

in which

91

e" = - A/r4

D" = - B

92

APPENDIX - II

Response Evaluation by SRSS for Nonproportionally Damped Systems

The system damping matrix constructed from element damping matrices

will usually be nonclassical or nonproportional. That is matrix [~]T[C]

[¢] will have off diagonal terms. Analysis is, however, considerably

simplified if these off-diagonal terms are neglected. To examine what

effect this ommission of off-diagonal terms will have on the system

response, a method has been developed in this investigation to obtain

exact response for the case where off-diagonal terms are included. This

exact method is described in the paper given in this appendix. The

paper will appear in the December 1980 issue of the Journal of Engineer­

ing Mechanics Division, ASCE (46).

The method is cast in the form of the conventional ~quare-£oot-of­

the-~um-of-the-squares (SRSS) procedure, and it can use a given set of

response spectra curves directly as seismic design input.

Some response results for the layered soil system being considered

in this investigation are obtained by this newly developed approach and

compared with the results obtained by the approximate approach in which

the off-diagonal terms are ignored. Tables 2 and 3 show these results.

The error is of the order of 0.35 percent. This small magnitude of

error in the response justified the use 6f the approximate approach, at

least for the soil systems being examined in this investigation.

93

SEISMIC RESPONSE BY SRSS FOR NONPROPORTIONAL DAMPING

by

1Mahendra P. Singh, M. ASCE

Introduction

Seismic analysis of structures with mass and (or) stiffness propor-

tional damping matrix can be conveniently performed by normal mode

approach. Especially, for obtaining the design response for a design

input prescribed in the form of ground response spectra, the method of,

so-called, square-£oot-of-the-sum-of-the-squares (SRSS) or modified SRSS

can be used for such a special damping case. Theoretical basis of using

SRSS to combine the maximum modal responses to obtain the design re-

sponse is fairly well established now. See references 1, 17 and 19.

The difficulty arises when the damping matrix of the system is not

proportional to either mass and (or) stiffness matrix or is not of a

special form as discussed by Caughey (3). Such a damping is usually

called nonproportional or nonclassical. The normal modes (undamped

modes) of the system then cannot be used to decouple the equations of

motion. Cases where the damping matrix of the system are nonpropor-

tional are· encountered rather quite often (24). Whenever the structural

model to be analyzed consists of elements made of different material

with significantly different energy dissipation characteristics, the

total damping matrix assembled from individual elemental damping ma-

trices are usually non-proportional. Such cases occur whenever combined

soil and structure analytical models are used to incorporate soil-

lAssociate Professor, Dept. of Engineering Science &Mechanics, VirginiaPolytechnic Institute & State University, Blacksburg, Virginia.

94

structure interaction effects in the dynamic analysis (4, 5, 18). Also

in the equivalent linear analysis of earth structures with strain­

dependent soil properties, the total damping matrix of the system will

usually be nonproportional.

Since the damping matrix can not be decoupled in such cases, very

often time-history analysis using numerical integration techniques are

used if the time history of the earthquake input is known. In some

cases undamped normal modes can still be used with advantage for such

time history analyses (4). However, to obtain design response for a

prescribed seismic design input like ground response spectra, the time

history approach becomes rather impractical and expensive since an

ensemble of time histories representing all possible ground motion must

be considered to obtain a reliable estimate of the design response.

Other approximate approac~es have ~een proposed where normal modes

can still be used. In the most commonly used approach, the off-diagonal

terms representing modal coupling through damping matrix are completely

ignored (2, 18) and diagonal elements of the (transformed) matrix are

used to obtain the modal damping ratios. In other approaches, the modal

damping ratios are obtained based on some weighting schemes (13, 18, 25)

or other techniques (6, 21, 22).' These modal damping ratios are then

used in the decoupled modal equations for time history analysis or with

the method of SRSS to obtain the design response. The rationale behind

the development of some of these simplified procedures appear to be

reasonable. Yet, however, these approaches are subjective and approxi­

mate for a given general damping case. The results substantiating

(and/or contradicting) various proposed approaches have appeared in the

95

literature. See References 2 and 18 on one side and References 4 and 7

on the other. The reported contradictions are probably due to the

choice of different example problems used for illustrations. Thus, in

general, the applicability of various proposed approaches to all pos­

sible cases of systems and responses cannot possibly be verified and

claimed.

For the purpose of obtaining design response for inputs prescribed

in the forms of ground spectra, a methcd is developed in this paper in

which nonproportional damping matrix can be used in essentially an SRSS

type of approach. For. this, first a formulation for the analysis of a

nonproportional damping system is developed for seismic inputs char­

acterized by power spectral density functions or other random forms.

This formulation is then extended for its use with inputs prescribed by

ground response spectra. A similar extension of the random vibration

formulation is made in the proportional damping case to justify the use

of SRSS for modal response combinations (1, 17, 19). The proposed

approach is, therefore, a form of modified conventional SRSS approach

where nonproportional damping effects can be included properly.

Analytical Formulation

The equations of motion of a multi-degree-of-freedom structure sub­

jected to a base excitation due to seismic ground disturbances can be

written in the following matrix form:

[M]{i} + (C]{~} + (K]{x} = - [M]{r}Xg

(1)

in which [M] = mass matrix, [C] = damping matrix not necessarily propor­

tional, [K] = stiffness matrix, {x} = vector of relative displacement

with respect to the base, {r} = influence coefficient vector (5), and

Xg(t) = base acceleration time history. The dots over a vector repre-

96

sent its time derivative.

If [C] is nonproportional or nonclassical (3), Eqs. 1 can not be

decoupled using the normal modes. For a correct modal analysis of such

system, a 2n-dimensional state vector approach is commonly used. See,

for example, the paper by Foss (8) and the texts by Frazer, Duncan and

Collar (9), Hurty and Rubinstein (10) and Meirovitch (15). This state

vector approach has been used by Itoh (11) for time history analysis of

a nuclear power plant system with nonproportional damping.

In the state vector approach, Eq. 1 is cast into a 2n-dimensional

equations (with the help of an identity equation, Ref. 9) of the follow-

ing form:

(A]{y} + [B]{y} = - (D]{~} Xg

in which

[[OJ [Ml] [-[Ml [OJ][A] = B =

[M] [C] [0] [K]

= ~[Ol [OJ][D]

[0] [M]

(2)

(3)

are 2n x 2n dimension matrices (n = number of generalized degrees-of-

freedom of the system), and {y} is a 2n x 1 dimension vector as follows

-_ f{{xx·}l'{y} L J (4)

Solution of Eq. 2 can be obtained in terms of eigenvalues and eigenvec-

tors of its homogeneous equations. This requires solution of the fol-

lowing 2n x 2n dimensional eigenvalue problem:

p[A][~] + [B][~] = {a}

where p = eigenvalue and (~J = 2n x 2n matrix of eigenvectors

97

(5)

{¢. }.J

Solution of this eigenvalue problem will give 2n-complex eigen-

values and a corresponding number of eigenvectors. Complex values are

accompanied by their conjugates. Some good algorithms are available for

solution of such eigenvalue problems. Required subroutine programs can

be found on most computing systems, such as in IMSL Library (12).

These eigenvectors are orthogonal with respect to matrices [A] and

[B]. Using the expansion theorem with the following transformation:

(6)

and orthogonal properties of the eigenvectors with respect to [A] and

[B] matrices, Eqs. 1 can be transformed into a decoupled set of equa-

tions as follows:

(7)

where [A*] = [~]'[A][~] and [B] = [~]'[B][~] are diagonal matrices, with

their diagonal elements related as follows:

A*. = - p. B*.J J J

h . h .th. 1 f E 5were p. ~s t e J-- e~genva ue 0 q. •J

(8)

Herein, a prime (') over a

matrix or vector represents its transpose.

A decoupled jth equation of Eqs. 7 is, therefore,

z. - p.z. = F.X (t)J J J J g

(9)

where F. is an element of complex vector {F} defined as follows:J

{F} =- [A*]-l[~]'[D]{Q} (10)r

The solution of Eq. 9 can be written as

t .. p.(t-T)z.(t) = F. J X (T)e J dT (11)J Jog

wherefrom, the solution for {y} is obtained from Eq. 6. The lower half

of this vector represents the relative displacement {x}. Thus using

only the lower half of modal matrix [~], the vector {x} can be written

as

98

2n{x} = l. {cp j h jj=l

2n t .. P (t-T){x} = l. {CPj}Fj J x (T)e j dT

j==l 0 g

2n t .. p. (t-T){x} == I {qj} J x (T)e J dT

j=l 0 g

(12a)

(12b)

(12c)

in which {cp.} the jth complex mode shape (lower part only) and {q.} =J J

F. {cp •}.J J

The p.s, which are complex eigenvalues, will have a negative realJ ,

part for a well supported stable structural system. If a complex eigen-

value analysis of a system with proportional damping matrix is per-

formed, the real and imaginary parts, ;. and e., respectively, of anJ J

eigenvalue will be related to the natura~ frequency and damping ratio

as follows

p. = - ;. + ie.J J J

::= - S.w. + w./l-S~ i (13)J J J J

where w. ::= jth natural frequency, S. = damping ratio (ratio of actualJ J

damping to the critical damping) in the jth mode and i =;.:r. Thus the'

imaginary part defines the damped natural frequency of the system and

the real part describes the decay characteristics of the response as

governed by the damping ratio S..J

The same interpretation can also be used to characterize the real

and imaginary parts of a particular eigenvalue p. in a nonproportionalJ

damping case, as was done by Minami and Sakurai (16) in their investi-

gat ion of soil-structure interaction effect.

solution of real and imaginary parts as

99

Thus, w. obtained from theJ

W.=~.1:+8.2J J J

are the undamped natural frequencies of the system.

S. = ~./1~.2 +8. 2J J J J

(14)

Also S. obtained asJ

(15)

could be interpreted to represent the modal damping ratio. This reali-

zation helps in transforming the response obtained from Eq. 12 in terms

of the modal response.

For the ground motion, X (t), being characterized by a random pro­g

cess the autocorrelation characteristics of any response x (t) can beu

obtained as follows

Zn 2nI I q.(u)qk(u)

j=l k=l J

(16)

in which

For a zero mean process, Eq. 16 also gives the covariance function of

the response process x (t). For a given description of the autocorrela­u

tion function E[Xg(Tl)Xg(TZ)]' Eq. 16 could possibly be evaluated to

obtain the covariance of the response. If the ground motion can be

assumed to be represented by a stationary 'random process with a spectral

density function, ¢ (w), Eq. 16 can be written asg

2n 2n cot1

t zE{x (tl)x (tZ)} = L L q·qk f ¢ (w) f f expliw(Tl-TZ)

u u j=l k=l J -00g 0 0

+ Pj(tl - T1) + Pk(tZ - TZ)]dTldTZdW (17)

q.(u) has been replaced by q .• Henceforth the same notationJ J

will be adopted in the paper.

To put this equation in the form so that the modal response can be

easily identified, the summation over 2n terms is represented as a

summation over n terms in which complex conjugate terms are considered

100

(20)

its real and imaginary parts as in

in pairs as follows:

P~(t1-Tl) Pk(tZ-TZ) p~(tZ-TZ)+ qje J ] [qke + q~e ]dTldTZdw (18)

where asterisk (*) as a superscript represents the complex conjugate of

a quantity. To further simplify this equation and put it in a more

usuable form, the terms with j=k and jfk (called cross terms here) will ~

now be evaluated separately.

A term with j=k, represented here as R.~ consists of four terms asJ

follows:

00 t l t z (Rj = [-00 ~g(w)Io f

oq~exP{iW(Tl - TZ) + Pj(tl - Tl ) + Pj(tz -'TZ)}

+ q~Z exp{iW(Tl - TZ) + P~(tl - Tl ) + p~(tz - TZ)}J ' J J

+ QjQj[exp{iw('l - 'Z) + Pj (t1 - 'Ii + Pj(tz - ·,Z)}

+ exp{iw('l - 'Z) + Pj(t1 - '1) + Pk(tz - 'Z) ~ )d'ld'zdw (19)

The integrals over Tl and TZ can be carried out separately with W

as a parameter. For the situation of stationary response, i.e. when t l

~ 00, t z ~ 00, but (tl

- tZ

) + T, the integrals in Eq. 19 are obtained as .

follows:

foo iWT[ q~ qj2 * [ 1R.(T) = ~ (w)e 2 Z + --~Z~~Z- + qJ.qJ. (-p.+iw)(-p~-iW)

J -00 g (p +w) (p~ +w ) J Jj J

+ {-Pj+iw~ (-PFiw) ] dw

With p. expressed in terms ofJ

Eq. 13, and q. asJ

101

q. = a. + ib.J J J

(21)

and combining separately the 1st and Znd parts and the last two parts in

the parenthesis in Eq. 20 and after some simplification, this equation

can be put into a recognizable form of integrations on Was follows:

00

J 2 2 2 I 12 iWTR.(T) = ~ (w)(4a.w + 4A.w.) H.(w) e dwJ _00 g J J J J

in which

2 2 2 2 ~A. = b. + (a. - b.)S. - 2a.b.S.vl-S~

J J J J J J J J J

(22)

(23)

and Hj(W) is the well known frequency transfer function of a single­

degree-of-freedom oscillator defined as

Similarity

1H. (w) =--,:--~~---

J (w:-w2)+2iB.w.wJ J J

of Eq. 22 with the

(24)

one encountered in the case of a propor-

tional damping matrix (19) may be noted.

Algebraic manipulations of the integrals in the cross terms with

j~k are more involved but similar. Here again, a cross term will con-

sist of four terms as follows:

co

Jt l

~ (w) Jg 0

+ qjq~ eXP{Pj(tl-Tl )

+ qjqk exp{pj(tl-Tl )

+ P~(t2-T2)}

+ Pk (t2-'2) ~ d'ld'2dw (Z5)

These integrals over Tl

and TZ can be evaluated separately for each

term. For the stationary response case, i.e., when tl~ co, t z + co and

(tl-t

2) = T, the above expression can be written as

q~qk*]

(-pj+iW)(-p~-iW)

102

+ qjq~ qjqk ]<-PJ.+iW)(-pk*-iW) + (-p*+iw)(-p -iw) dw

j k(26)

Substituting qj' qk' Pj and Pk in terms of their real and imaginary

parts, and after some algebraic manipulations, Eq. 26 can be put into

its following form:

00

in which

iWT= f ¢ (w)e [X~k(w) + iYJ~k(w)]HJ'(W)~(W)dw-00 g J

(27)

(28)

Eq. 27 can be further transformed into the real forms of H.(w) andJ

~(W) as

(29)

in which

(30)

in which, in turn, Ujk and Vjk are defined as

(31)

Noting that Xjk = ~j and Yjk = - Ykj , the two cross terms Rjk(T)

and ~j(T) when combined together will eliminate Yjk from the final ex­

pression. Using Eqs. 22 and 29, the autocovariance function of the sta-

tionary response, E[x (t+T)X (t)], can be obtained. For T = 0, it givesu u

103

the mean square response as follows:

(32)

where Aj and Xjk(W) are given by Eqs. 23 and 30, respectively. Again

similarity of this expression, with an equivalent response expression

for a proportional damping case (19) may be noted. This expression can

be used to obtain the response for an input prescribed in the form of

spectral density function. For commonly used Kanai-Tajimi type of

spectral density functions, the integrations in the above equation can

be routinely obtained by the method of residues. For the degitized form

of spectral density functions also, where variation between any contin-

guous discrete points can be assumed to be linear, the integrations in

Eq. 32 are obtained without much difficulty.

Design Response by SRSS

Eq. 32 represents a random vibration form of the (modified) square-

root-of-the-sum-of-the-squares approach for obtaining the root mean

square response. The terms under single summation represent the con-

tributions from individual modes whereas the terms under double summa-

tion represent the interaction effect between the modes. To obtain the

design response, the mean square response obtained from Eq. 32 should be

amplified by an appropriate value of the peak factor. The peak factor

could depend upon an acceptable level of probability of exceedance.

For a seismic input defined in the form of ground response spectra

curves, which define maximum (or design) response in each mode, the

modal peak factors are implicitly defined. Assuming that the peak fac-

104

tor is a constant value for each mode, and following the development

given in Ref. 19, Eq. 32 can be used to obtain the design response in

terms of modal response spectra values, as follows (also see Appendix I)

R2

(u) =x

n 2 2L 4[aJ.F (w

J,) + A

J.lI1 (w

J,) /W

J.

j=l

n n+ 2 L I {[Q'/r2 + F(W.)P']I1 (W.)/w7

j=l k=j+l J J J

+ [s' + F(Wk)R')II(wk)/w;} (33)

in which Rx(u) is the design response, r = wj/wk and pi, Q', R' and S'

are obtained from the solution of the following simultaneous equations

I a 1

u.1 s

v u t

v 0

(34)

where the terms u, v, s, t and WI' Wz' W3 and W4 are defined in Appendix

I. The values of I1

(Wj

) and F(Wj), as discussed in Ref. ZO, are defined

in terms of frequency integrals as follows.

F(w.) = IZ(W.)!Il(W.)J -J J

(35a)

(35b)

(36)

in which Cf

is the peak factor and IZ(Wj

) is related to the relative

velocity response of the oscillator of frequency w. and is defined asJ

co

IZ(W') = c2f

J w:w2~ (W)!H.(W) 12

dWJ -eoJ g J

The characteristics of I 1 (Wj), I 2 (W

j) and F(W

j) are discussed in

Ref. 20. For a Newmark type of smooth ground response spectra, Ref. 23,

these terms may be defined by the following equations.

105

2R (w.)J (37)

(38)

=

response spectrum value at frequency w. andJ

= the control frequency at which the ground spectra

in which R(w.) = groundJ

damping ratio 6j

, w~

starts to drop (such as 9 cps in R.G.-l.60 NRC spectra, Ref. 23), w. u

the frequency beyond which there is no spectral amplification of the

ground motion (such as 33 cps for the NRC spectra), and A is the maxi­g

mum ground acceleration.

Eq. 33 defines the form of SRSS which should be used to combine

modal responses to obtain the design response in a nonproportiona1

damping case. This equation is similar to the modified SRSS equation

(19) which is used in a proportional damping case. In the latter case,

the contribution of the double summation terms (also called cross terms)

can be neglected if the frequencies are far apart, i.e., when ratio r =

Wj/wk is significantly different from 1.0. In such a case, the correla­

tion between modes is insignificant. Such a simplified statement,

however, cannot be made in the case of nonproportiona1 damping. As can

be seen from the cross terms in Eqs. 32 or 33, this correlation is

affected not only by the closeness of frequencies but also by the phase-

lag relationship between the modes. This effect is brought in through

the real and imaginary components, a. and b., of the complex modes. InJ J

general, therefore, for a nonproportiona1 damping case the cross terms

should be considered in the analysis. The form of Eq. 33, however,

should not present any special difficulty in doing so in an analysis

procedure.

The design response of any other quantity of interest, such as

106

member forces, stress, story shear, etc., can also be obtained, {f ap-

propriate complex modal response values are used in Eqs. 12 in place of

{¢.}'s. For a linear structure, such response quantities are linearlyJ

related to displacement mode shapes through the stiffness matrix [K].

However, the elastic and the inertia forces now are not related as sim-

ply as they are in the case of proportional damping, Ref. 5. Thus the

use of easy-to-use diagonal mass matrix [M], therefore, cannot be made

to obtain the elastic force response.

In some engineering structures, the higher modes do not contribute

much to the total response, and, therefore, in the proportional damping

case the summation to obtain the response is usually performed only over

a first few modes. Extension of this procedure to a nonproportional

damping case appears to be acceptable but may have to be verified by

calculations involving increasing number of modes.

Numerical Results

To verify Eq. 32 (and thus Eq. 33 also), the response results for a

system with proportional damping were obtained by the method proposed in

this paper, Eq. 32, and by the conventional normal mode approach (which

also gives mathematically exact response in this case). A Kanai-Tajimi

type, power spectral density function was used as a seismic input. The

response results obtained by the two approaches were the same, which

numerically verified the validity of Eq. 32.

As an example of a nonproportional damping case, an artificial

problem of a three story building with a spring and dashpot to represent

the soil half space effect was analyzed. The elements of the mass,

stiffness and damping matrices used are given in Appendix II. Here the

107

assumed damping matrix could not be diagonalized by undamped normal mode

shapes of the structure. The exact root mean square response results

obtained for displacements, elastic forces and the story shears from Eq.

32 are shown in Table 1. The seismic input in the form of Kanai-Tajimi

type of power spectral density function, as defined in Ref. 19, has been

used. Corresponding results obtained by the approximate normal mode

approach in which off-diagonal terms are neglected are also shown in the

table for comparison. The magnitude of the percent error in the values

obtained by approximate undamped normal mode approach may be noted.

This difference will, of course, depend upon the parameters of the

system and the type of response obtained and thus should not be con­

sidered typical. Larger differences can possibly be obtained for

systems where the modal frequencies are close and modes of the response

quantity interact with each other significantly. No further attempt was

made to construct systems where more severe differences in the approxi­

mate and exact solutions could be shown. However, such cases can

possibly be encountered in many practical situations, and the use of the

method proposed in this paper can be made to obtain accurate response

results. Also, no attempt was made to evaluate various approximate

procedures described in the literature on the subject vis-a-vis the

exact procedure presented here; such an evaluation may not be conclusive

as it will be strongly affected by the structural model and response

quantities chosen in the evaluation process.

Summary of Proposed Procedure

For the benefit of an analyst who may not wish to go through the

details of the formulation given in the paper, the steps required for

108

the implementation of the proposed SRSS procedure to obtain the design

response for a prescribed ground spectra are outlined below.

Step 1: Obtain the complex eigenvalues and eigenvectors of the 2n

x 2n dimensional eigenvalue problem defined by Eq. 5. As these values

occur in complex conjugate pairs, consider only one of them. Only a

first few sets may have to be considered.

Step 2:- Obtain modal frequencies, Wj' and damping ratios, Sj'

using Eqs. 14 and 15 for each eigenvalue selected in Step 1.

Step 3:

ing Eq. 10.

Obtain the complex values of F. for each selected mode us­J

Step 4: Obtain the eigenvectors of the response quantity of inter-

est by appropriate linear transformation relating the response quantity

with the relative displacement modes {~.}, e.g.J

{g.} = [T]{~.} (39)-J .J

in which {g.} = the jth modal vector of the response quantity and [T] isJ

the appropriate transformation matrix. Only lower half of the eigen-

vectors which represent the relative displacement quantities should be

considered.

Step 5: Obtain {qj} = Fj{gj} and obtain the real and imaginary

parts of {q.} ={a. + ib.}.J J J

Step 6: Obtain Il(Wj

) and F(Wj ) using Eqs. 35, 37 and 38 corres-

ponding to each modal frequency w. and damping ratio S..J J

Step 7: For each pair of cross terms obtain pi, Q', R' and Sl from

the solution of Eqs. 34 with the definitions of various terms therein

given in Eqs. 43 of Appendix I.

Step 8: Sum up the modal responses according to Eq. 33 to obtain

the total design response.

109

Summary and Remarks

A modified SRSS approach is presented to obtain the seismic design

response of systems with nonproportional damping. Nonproportional damp­

ing effects are included exactly in a mathematical sense in the proposed

approach. Seismic input defined in the form of ground response spectra

or spectral density function can be conveniently used. To cast the

procedure in the form of SRSS, the assumption of stationarity of earth­

quake inputs is made. Since the earthquake motions are nonstationary in

character, this assumption is a point for criticism. However, the

analytical basis of the conventional method of SRSS or its other modified

forms, which provide acceptable and reliable estimates of design res­

ponse of proportional systems, is also the same, and the validity of

such a basis has been verified by numerical simulation studies. In this

regard, therefore, the use of the SRSS proposed here for nonproportional

systems should be as accurate as the'conventional SRSS for proportional

systems. Furthermore, the use of response spectrum values in Eq. 33

does introduce the nonstationary effect as these values are obtained

from prescribed ground spectra which are generated from ensemble of

recorded accelerrograms.

The proposed approach requires the solution of a complex eigenvalue

problem. Also, the size of the eigenvalue problem is twice as large as

the size for a proportional damping case. This obviously means a sig­

nificant increase in computational expense, especially if the system is

large. However, efficient and accurate computational algorithms of such

eigenvalue problems are now available on most computing systems.

Therefore, accessibility of such algorithms and a desire to obtain a

more analytically accurate design response should justify additional

110

computational cost. More accurate response can also be obtained by time

history analysis of a smaller size problem. However, a large number of

such calculations with a large ensemble of time histories as input may

be necessary to establish a credible value of the design response. This

may more than offset the computational cost in the favor of the proposed

SRSS scheme.

Acknowledgements

The assistance provided by M. Ghafory-Ashtiany, research assistant at

VPI & SU, in numerical verification of the proposed approach is ack­

nowledged. This work is supported by the National Science Foundation

through Grant No. PFR-7823095, with William W. Hakala as its Program

Manager. This support is gratefully acknowledged. Any opinions, find­

ings and conclusions or recommendations expressed in this publication

are those of the writer and do not necessarily reflect the views of

the National Science Foundation.

111

APPENDIX I - DESIGN RESPONSE

The design response corresponding to the mean square response

defined by Eq. 32 for a peak factor value of Cf

can be written as fol­

lows:

n co

R2(u) C2 L f 2 2 2 I 2= 4(a.w + A.w.) H. (w) I <P (w)dwx f

j=l -co J J J J g,

n n co

+ 2C2 I L f x·k(w)IH.(w)121~(w)12<p (w)dw (40)f

j=l k=j+l -co]] g

Assuming that the peak factor relating an oscillator root mean square

response to the ground response spectrum value is also Cf , a term under

single summation of Eq. 32, in view of Eqs. 35 and 36, can be written as00

4C2 f (a7w2 + A.(7) !Hj(W) !2<p (w)dwf _00] ] ] g

.2 2=4(a.F(w.) + A.)Il(W.)/w.J J J J J

(41)

To express a cross term into the form given in "Eq. 33, the inte-

grand is resolved into partial fractions as,

P'w2 + Q'w

2k

== -2:=---:2:--=2--::2:-:::2~2 +(w.-w ) +413.w.w

J J J

(42)

in which P', Q', R' and S' are obtained from solution of four simu1-

taneous Eqs. 34, with various terms therein defined as follows:

224s = - 2r (1 - 213.), t == r

]

2u == - 2(1 - 2I3

k), v = 1

(43)

112

The integrals of these partial fractions can be defined in terms of

response spectrum values with the help of Eq. 35 and 36 to give the

double summation term of Eq. 33.

Refinements in which the peak factor Cf is considered to depend

upon the frequency w. can also be incorporated in this formulation;J

however, they are considered unnecessary unless the design response

quantity is strongly affected by the higher frequency modes. See Ref.

12.

113

APPENDIX II - MASS, STIFFNESS AND DAMPING MATRICES OF EXAMPLE PROBLEM

The elements of the mass, stiffness and damping matrices used in

the example problem are given as follows:

300 0 0 0

0 24.0 0 0kips-sec2/ft[M] =

0 0 18.0 0

0 0 0 12.0

38160 -2160 o o

[K] =-2160 3600 -1440 0

o -1440 2160 -720

o 0 -720 720

kips/ft.

The damping matrix was constructed from the fixed base structural

damping matrix which had 5% damping in all the modes and a soil damper

with damping constant e = 1050 kips-sec/ft. This resulted into a com­

bined damping matrix as shown below

1081.14 -20.73 -7.01 -3.39

[e] =:

-20.73 28.32 -6.73 -0.85

-7.01 -6.73 18.07 -4.33

-3.39 -.85 -4.33 8.58

114

kips-sec/ft.

APPENDIX III - REFERENCES*

1. Amin, M. and Gungor, 1., "Random Vibration in Seismic Analysis ­An Evaluation", Proceedings, ASCE National Meeting on StructuralEngineering, Baltimore, MD, April 19-23, 1971.

2. Bailak, J., "Modal Analysis for Building - Soil Interaction",Instituto de Ingenieria, Universided Nacional Autonoma de Mexico,1975.

3. Caughey, T. K., "Classical Normal Modes in Damped Linear DynamicSystems", Journal of Applied Mechanics, Vol. 27, June 1960, pp.269-271.

4. Clough, R. W. and Mojtahedi, S., "Earthquake Response Analysis Con­sidering Non-Proportional Damping", Journal of Earthquake Engineer­ing and Structural Dynamics, Vol. 4, pp. 489-496 (1976).

5. Clough, R. W. and Penzien, J., Dynamics of Structures, McGrawHill, 1975.

6. Cronin, D. L., "Approximation for Determining Harmonically ExcitedResponse of Nonclassically Damped Systems", Journal of Engineeringfor Industry, Transactions of the ASME, 98B, pp. 43-47 (1976).

7. Duncan, P. E. and Taylor, R. E., "A Note on the Dynamic Analysisof Non-Proportionally Damped Systems", Journal of Earthquake En­gineering and Structural Dynamics, Vol. 7, pp. 99-105 (1979).

8. Foss, K. A., "Co-ordinates which Uncouple the Equations of Motionof Damped Linear Dynamic Systems", Journal of Applied Mechanics,Vol. 25, No.3, September 1958, pp. 361-364.

9. Frazer, R. A., Duncan, W. J. and Collar, A. R., Elementary Ma­trices, Cambridge University Press, 1965.

10. Hurty, W. C. and Rubinstein, M. F., Dynamics of Structures, Pren­tice Hall, Inc., Englewood Cliffs, N.J., 1964.

11. Itoh, Tetsuji, "Damped Vibration Mode Superposition Method forDynamic Response Analysis", Journal of Earthquake Engineering andStructural Dynamics, Vol. 2, pp. 47-57 (1973).

12. IMSL Library, "International Mathematics and Statistics Library",IMSL LIB 0007, IMSL Inc., Rev. 1979.

13. Johnson, T. E. and McCaffery, R. J., "Current Techniques forAnalyzing Structures and Equipment for Seismic Effects", paresentedat the ASCE National Meeting on Water Resources, New Orleans, LA"Feb. 3-7, 1969.

14. Kiureghian, Armen Der, "Probabilistic Response Spectral Analysis",3rd ASCE/Engineering Mechanics Division Specialty Conference, Uni­versity of Texas, Austin, TX, Sept. 17-19, 1979.

* These references are only for the paper in Appendix II.

115

15. Meirovitch, L., Analytical Methods in Vibration, Macmillan Com­pany, New York, 1967.

16. Minami, J. K. and Sakurai, Jo, "Earthquake Resistant Design ofBuildings Considering Soil Types and Foundation Construction",Memoirs of the School of Science & Engineering, Waseda Univ., Noo41, 1977, Reprinted in the Newsletter, EERI, Vol. 13, No.5, PartB, Sept. 1979.

17. Newmark, N. M. and Rosenblueth, E., Fundamentals of EarthquakeEngineering, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1971.

18. Rossett, J. M., Whittman R. V. and Dobry, Ro, "Modal Analysis forStructures with Foundation Interaction", Journal of the Structural ­Division, ASCE, Vol. 99, No. ST3, pp. 399-416, 1973.

19. Singh, M. P. and Chu, S. L., "Stochastic Considerations in SeismicAnalysis of Structures", Journal of Earthquake Engineering andStructural Dynamics, Vol. 4, pp. 295-307, 1976.

20. Singh, M. P., "Seismic Design Input for Secondary Structures",Journal of Structural Division, ASCE, Vol. 106, No. ST2, ProceedingPaper No. 15207, February 1980, pp. 505-517.

21. Thomson, W. T., Calkins, T. and Caravani, P., IIA Numerical Studyof Damping", Journal of Earthquake Engineering and Structural Dy­namics, Vol. '3, pp. 97-103 (1974).

22. Tsai, N. C., "Modal Damping for Soil-Structure Interaction", Jour­nal of Engineering Mechanics Division, ASCE, EM4, April 1974, pp.323-341.

23. u.s. Nuclear Regulatory Connnission, "Design Response Spectra forNuclear Power Plants", Nuclear Regulatory Guide No. 1.60, Washing­ton, D.C., 1975.

24. Warburton, G. B. and Soni, S. R., "Errors in Response Calculationsfor Non-Classically Damped Structures l1

, Journal of Earthquake En­gineering and Structural Dynamics, Vol. 5, pp. 365-376 (1977).

25. Whitman, R. V., "Soil-Structure Interaction", Seismic Design forNuclear Power Plant, R. J. Hansen, ed., MIT Press, Cambridge,Mass., 1970, pp. 241-269.

116

APPENDIX IV - NOTATIONS

a. ,b.J J

Ag

A.J

[A],[B]

c

D1,D2

E2,E3

F(w.)J

{gj}

H. (w)J

i

II (wj),I2 (Wj )

[K]

[M]

P',Q',R',S'

r

s,t,u and v

t l ,t2

Ujk and Vjk

- real and imaginary parts of q. defined in Eq. 21J

- maximum ground acceleration due to gravity

- a factor as defined in Eq. 23

2n x 2n matrix as defined in Eq. 3

- damping constant

factors defined in Eq. 43

- peak factor

- damping matrix

factors defined in Eq. 43

factors defined in Eq. 43

- factor defined in Eq. 35

- vector for a response quantity defined in Eq. 39

- relative displacement frequency transfer functionEq. 24

- complex number ;.:r

- factors defined by Eqs. 34 and 36 or 37 and 38

- stiffness matrix

- mass matrix

- complex eigenvalue of Eq. 5

- factors defined by Eq. 34

- uth element of vector {qj}

- jth element of a response quantity, defined by Eq. 39

- frequency ratio = Wj/Wk

- defined by Eqs. 43 in Appendix I

- time variables

- defined by Eqs. 31

117

x (t)u

X (t)g

Xjk(w), Yjk (w)

Xjk

(w), Yjk(W)

{y}

z.J

Bj

~j

{<t>j}

[<Ill

e.J

1"1'1"2

W

w.J

- relative displacement response of a point u

- ground acceleration

- factors defined by Eqs. 28

- factors defined by Eq. 30

- state vector, Eq. 4

- jth complex valued principal coordinate, Eq. 9

- damping ratio in the jth mode, Eq. 15

- real part of Pj

- jth relative displacement mode shape

- modal matrix

- complex part of p.J

- dummy time variable

- frequency variable

- jth mode frequency

118

Table 1 - Root-Mean Square Response of the Structural System with Mass,Stiffness and Damping Matrices as shown in Appendix II

Displacement Elastic Forces Story Shearsin ft-units kips kips

Node or Exact Approx. Percent Exact Approx. Percent Exact Approx. PercentSpring Approach Approach Ditf. Approach Approach Ditf. Approach Approach Ditf.

No. Eq. 32 xlO-3 10-3 x10-3 x10-3

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)

1 0.1700 .1802 5.9 50983 6.370 6.5 6.121 6.488 600

2 .6367 .6623 4.0 .784 .915 16.7 1.250 1.305 4.4

3- 1.1448 1.1489 0.3 .714 .735 2.8 .880 .887 0.8

4 1. 7411 1. 7657 1.4 .670 .724 8.1 .670 .724 8.1

119

Abstract

A method is presented to obtain seismic design response of linearly

behaving structures with nonproportional damping characteristics. For

such systems, time history analyses are usually performed to obtain

accurate seismic response. To obtain design response, such procedures

can become expensive and cumbersome as an ensemble of time histories may

have to be considered as seismic input. Other procedures have been

developed to ascertain appropriate values of modal dampings so that

modal analysis approach and thus commonly adopted square-~oot-of-the­

sum~of-the-squares (SRSS) procedures can be used. These approaches are,

however, approximate and numerical results in favor as well as against

the use of these approximate procedures have appeared in the literature.

The approach presented here considers nonproportional damping effects

exactly in analytical sense. It uses the state-vector formulation and

is based on random vibration principles. In its final form the approach

is similar to conventional SRSS approach, and thus ground response

spectrum can be directly used to obtain a design respose. Details of

the proposed procedure are outlined for its direct implementation by a

potential user.

120

Civil Engineering Abstract

A modal analysis method which can include nonproportional damping

effects accurately is presented to obtain seismic design response of

linearly behaving structures. The format of the proposed method is the

same as that of the conventional response spectrum SRSS approach. Thus

prescribed design response spectra can be used directly to obtain a

design response.

121

Key Words - Earthquakes, Seismic Design Response, Response Spectra,

Dynamic Analysis, Damping, Nonproportional Damping, Modal

Analysis, Structural Analysis, Dynamics, Vibrations

122

APPENDIX III

2Expected Values E{G (s)x x } and E[G (s)x ]:q r s q r

It is assumed that E, x and x are jointly and individually normalr s

random variables. If the excitation is a Gaussian random process, the

response of the equivalent linear system represented by Eq. 2 in a give

iteration will.also be normal. In such a case the assumption of normal-

ity for s, x and x will be valid. The expected value E{G (s)x x }r s q r s

can be written in terms of the joint probability density function of the

variables as follows

E{G (E)X x } = [f [ G (s)x x f (x ,x ,E)dx dx dsq r s q r s r s r s

s x xr s

= f G (E){[ f x x f (x ,x !s)dx dx }f(S)dss q x x r s r s r s

r s

(III .1)

= fs

G (E) E(x x Is) f (S)dEq r s

(III. 2)

For jointly normal variables with zero mean, the conditional mean values

and covariance can be written as follows (32):

cllx =

r

E(x s)E(x Is) = ~r~

r a;CE· ]J, x

a

E(x E)= E(x IE) = _.....:;s'--- s

s a~

= E[(x _]JC )(x _]Jc ) Is]r x s x .

r s(III. 3)

E(x E)E(x E)= E(x x ) _ r s

r s a~

In terms of these conditional mean values and the covariance, the condi-

tional expected value E(x x Is] can be written as follows:r s

E(x x Is] = cov(x ,x Is) + E(x IE)E(x IE)r s r s r s

E(x s)E(x s) E(x s)E(x s) 2r s r s= E(x x ) - -~~:---- + -~~-~- E:r s 2 4as as

123

(III. 4)

Substituting for this conditional expected value in Eq. 111.2 we obtain

E{G (s)x x } = A f G (s)f(s)ds + B f s2G (s)f(s)dsq r s s q s q

in which

A = E(x x ) - E(x s)E(x s)/02r s r s S

4B = E(x s)E(x s)/or s s

(III. 5)

(III. 6)

2The values of E(x x ), E(x E), 0c' etc. can be obtained using Eqs:r s r COo

21-23. For the strain dependent shear modulus curves, such as shown in

Fig. 4, the integrals in Eq. 111.5 are obtained by numerical integrations

2of the density function feE) over G (8) and 8 G (E). For a Gaussianq q

input, the density function of strain s is a zero mean Gaussian random

variable with density ~unction defined as

2 2f(s) = exp(- E /20s)/(Osl2'IT)

The expected value of G (8)X2 are obtained similarly:

q s

For jointly normal X and ss

(III. 7)

(III. 8)

2 2p 0+ __s_ £:2

o~(III. 9)

where p = correlation coefficient

= E(X s)/o °s s s

Thus:

2 2p Os 2

+ ---2-- E[E G (s)}o q

s

124

(IlLIG)

(IlL 11)